Research ArticleFast Phased Array Calibration by Power-Only MeasurementsTwice for Each Antenna Element

Guolong He , Xin Gao, and Hui Zhou

Beijing Institute of Tracking and Telecommunication Technology, Beijing 100094, China

Correspondence should be addressed to Guolong He; heguolong@alumni.sjtu.edu.cn

Received 1 October 2018; Revised 23 February 2019; Accepted 10 March 2019; Published 17 April 2019

Academic Editor: Chien-Jen Wang

Copyright © 2019 Guolong He et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper proposes a novel power-only measurement method for phased array antenna calibration. Besides the total array power,only one phase shift and two power measurements for each array element are required to determine the element complex electricfield distortion, one by shifting the element’s phase of π/2 and the other by turning the element under test off. The theory of theproposed calibration method is given, and the closed form formulation of the element amplitude and phase distortions isderived. From the mathematical point of view, it is the minimum required measurements that use two scalar values todetermine one complex vector. Numerical simulations and experiments are conducted to validate and demonstrate theeffectiveness of the proposed calibration method.

1. Introduction

Phased array antenna plays a very important role in modernradar and wireless communication systems [1–9]. Due tosome inevitable errors and uncertainties, such as mechanicalmanufacturing tolerance, temperature variation, and elec-tronic and radiofrequency component aging, the realisticarray element excitation (both amplitude and phase) deviatesfrom their ideal values. This would cause array radiationpattern distortion and degrade the overall performance,which must be carefully considered for practical systemdesign. Hence, to maintain an acceptable performance ofphased array antenna, calibrations are always necessaryby measuring the excitation distortions of individual arrayelements and compensating them as accurate as possiblebefore and/or during their operation.

A variety of calibration techniques have been proposed sofar for phased array antenna calibration [10–23]. In [10],Silverstein proposed a method which controls the elementphases based on time-multiplexed orthogonal codes; thus,the individual element excitation can be derived from thecombined array signal. Some other phase toggling methodschange the element’s phase shift to gather the requiredelement excitation information [11–14]. Lee et al. [11]

switched the element’s phase between the two states of0 and π and determined the element excitation by comparingthe combined array signal at these two states. For all theabovementioned methods, they require coherent measure-ment between the calibration source and sink to obtain thecomplex value containing both amplitude and phase infor-mation. It is not always convenient to apply them for operat-ing phased array system, especially for far-field calibrationscenario with a long distance between the source and sinkor in the millimeter wave band where the phase is sensitiveand can be measured only with a restricted accuracy.

Compared to these complex measurement methods,some other methods which involve amplitude- (or power)only measurements are considered to be more robust andconvenient. In such case, coherent measurements are nolonger required with precise synchronization between thetransmitter and receiver. Many efforts have also been madethat follow this principle. He et al. [15] proposed a methodby periodically modulating the element phase sequentiallyand then applying a fast Fourier transform (FFT) analysis;the element complex electric field excitation is obtained fromthe corresponding modulation frequency harmonics of thecombined array signal. The rotating element electric fieldvector (REV) method is another well-known power-only

HindawiInternational Journal of Antennas and PropagationVolume 2019, Article ID 6432149, 10 pageshttps://doi.org/10.1155/2019/6432149

measurement method; it measures the amplitude of the arrayby shifting the element phase from 0 to 2π continuously[16, 17]. The element complex electric field can be deter-mined from statistical signal processing of the threeparameters, the maximum and minimum power of the arraysignal, and the element rotation phase to maximum power.Although it is very simple and easy to use, however, a largeamount of measurements for each element is needed in theconventional REV method. In 2001, Sorace [18] pointedout that measurement results at four orthogonal phase shiftsare sufficient enough to obtain a maximum likelihood esti-mation for each array element. Long et al. [19] also derivedan expression that requires two phase shifts of π/2 and π toyield the element complex excitation information. All theseabove efforts are made to reduce the amount of requiredmeasurements and overcome the inefficiency of the originalREV method.

In this paper, we propose a novel power-only measure-ment method to calibrate the element complex excitationdistortion. For each array element, two power measurementsare enough to determine the element complex electric field,one by shifting the element’s phase of π/2 and the other byturning the element under test off. This method only needssome simple mathematical calculation but requires noauxiliary calibration element or any additional hardware.From the mathematical point of view, it is the minimumrequired measurements that use two scalar values to deter-mine one complex vector. The rest of this paper is organizedas follows. In Section 2, the proposed calibration method isintroduced, and the closed-form formulation for elementamplitude and phase distortions is derived. Then, in Section3, the calibration accuracy is analyzed mathematically; thestandard derivations of both the amplitude and phase estima-tions are obtained. Section 4 uses an eight-element patchantenna array to demonstrate the calibration procedure and

investigate its performance. In Section 4, a four-elementdipole antenna array is fabricated and tested to furthervalidate the effectiveness of the proposed method. Finally,this paper is concluded in Section 5.

2. Methodology

Figure 1 illustrates a typical configuration for N-elementactive phased antenna array. It contains both transmittingand receiving modes by adopting the T/R module. Eachantenna element has a phase shift for electronic beam scan-ning. The method proposed in this paper can be used innot only the transmitting mode but also the receiving mode.Let En and φn denote the amplitude and phase of the nthantenna element, n = 1, 2,…N . The phased array electricfield E, the combination of the individual element electricfields, can be described as follows:

E = E0 exp jϕ0 = 〠N

n=1En = 〠

N

n=1En exp jϕn , 1

where E0 and ϕ0 are the amplitude and phase of the arrayelectric field, respectively. As shown in Figure 2, in order tocalibrate the nth array element complex excitation distortion,we can divide equation (1) into two parts:

E0 = En + En = En exp jϕn + En exp jϕn= En cos ϕn + En cos ϕn + j En sin ϕn + En sin ϕn ,

2

where En = En exp jϕn denotes the combined electric fieldof the array expecting the nth element, and it has the power

T/Rmodule

Phaseshifter

PALNA

Beamformer

To transmitter/receiver

Figure 1: Typical architecture of an active phased antenna array.

2 International Journal of Antennas and Propagation

of Pn = E2n which can be easily measured by turning the nth

element under test off or terminating it with a matched load.From equation (2), the array power can be expressed as

P0 = E20 = En cos ϕn + En cos ϕn 2 + En sin ϕn + En sin ϕn

2

= E2n + E2

n + 2EnEn cos ϕn − ϕn

3

We denote the phase difference φn = ϕn − ϕn; thus,equation (3) can be further rewritten as

P0 = E2n + E2

n cos2φn + E2n sin2φn + 2EnEn cos φn

= En + En cos φn2 + En sin φn

2,4

where En cos φn and En sin φn are the real and imaginaryparts of the nth array element complex excitation,respectively.

We then shift the phase of the nth antenna element byπ/2; then, the corresponding electric filed Eπ/2 and the arraypower Pπ/2 can be written as

Eπ/2 = En exp jϕn + En exp jϕn exp jπ/2= En cos ϕn − En sin ϕn + j En sin ϕn + En cos ϕn ,

5

Pπ/2 = En cos ϕn − En sin ϕn2 + En sin ϕn + En cos ϕn 2

= E2n + E2

n + 2EnEn sin ϕn − ϕn= E2

n + E2n cos2φn + E2

n sin2φn + 2EnEn sin φn

= En + En sin φn2 + En cos φn

2

6

From equations (4) and (6), the two unknowns En cos φnand En sin φn can be derived as follows:

En cos φn =−2Pn + P0 − Pπ/2 +Q

4 Pn,

En sin φn =−2Pn − P0 + Pπ/2 +Q

4 Pn,

7a

or

En cos φn =−2Pn + P0 − Pπ/2 −Q

4 Pn,

En sin φn =−2Pn − P0 + Pπ/2 −Q

4 Pn,

7b

where

Q = 4PnP0 + 4PnPπ/2 + 2P0Pπ/2 − 4P2n − P2

0 − P2π/2 8

Since P0 is the same for all the elements, thus, onlytwo additional power-only measurements Pn and Pπ/2 arerequired to calculate each element excitation distortion.Like the conventional REV method, it also has two possiblesolutions in equations (7a) and (7b). This ambiguity can beeasily eliminated if the initial phases of the individualelements are set to make En > En [19]. From equations (7a)and (7b), the relative amplitude and phase distortion, knand xn, of the nth antenna element compared to the arraysignal can be derived as

kn exp −jχn = E0En

= En + En

En= En

Enexp jφn + 1, 9

where

En = En cos φn2 + En sin φn

2, 10a

φn = tan−1 En sin φn/En cos φn 10b

3. Calibration Accuracy Analysis

Recall that En cos φn and En sin φn are the real and imaginaryparts of the nth array element complex excitation, respec-tively. Since these two components are orthogonal to eachother, they can be modeled as independent Gaussian distri-butions with zero mean and the variance 1/ 2 × SNR , whereSNR is the signal-to-noise ratio of the power measurement.By squaring them and adding together as in equation (10a),the result of Pn should have doubly noncentral chi-squareddistribution with two degrees of freedom. From the abovederivation, the amplitude estimation En, as the square rootof Pn, should have a Rayleigh distribution, and it has thestandard deviation as [14]

σ En = 2SNR + 1SNR 11

The phase estimation φn, as shown in equation (10b),can be viewed as the ratio of two Gaussian distributions.

E

En

jEn

E0

E𝜋/2

O

Figure 2: Vector representation of the proposed calibrationmethod.

3International Journal of Antennas and Propagation

Its probability density function (PDF) can be expressed asfollows [24]:

p y = 1π y2 + 1 exp SNR + SNR

π y2 + 1 3/2

× exp −SNR ⋅ y2

y2 + 1 × erf SNRy2 + 1 ,

12

where y = tan φn also has approximate Gaussian distributionfor high SNR. Finally, the standard deviation of the phaseestimation in degree is approximated as [14]

σ φn ≈180

π 2SNR13

4. Numerical Simulations

In order to verify the effectiveness of the proposed method,an eight-element linear patch antenna array workingat S-band is numerically simulated to demonstrate thecalibration procedure and evaluate its performance. Thephased array contains eight half-wavelength spaced patch

antenna elements. As shown in Figure 3, the rectangularpatch antennas are with length h = 4 cm and width w =3 cm on the lossless substrate, which has the dielectricconstant εr = 2 2 and thickness of 1.57mm. The feed isoffset by 5.5mm from the center of the rectangular patchantenna.

During the simulation, random initial amplitude andphase excitation are preset for each antenna element. Thestandard deviation for element amplitude and phase erroris set to 0.5 dB and 10 deg, respectively. The power measure-ment is made at the far-field region of the phased array, andthe measurement SNR is set as 20 dB. Figure 4 illustrates oneexample of the simulated far-field power measurementresults. The black asterisks denote the normalized combinedarray power P0, which is normalized to 0 dB. The red andblue triangles denote the two power measurements Pn andPπ/2 for each array element. They vary from each other sinceevery element has different amplitudes and phase distortions.Figures 5 and 6 illustrate the calibration results for eachelement derived from the proposed method. Compared tothe preset values, the corresponding calibration errors are0.26 dB and 2.17 deg for amplitude and phase distortion,respectively. It can be seen that the estimation resultshave very good coincidence with the preset values. The

h

w

d

Figure 3: Geometry of the eight-element linear patch antenna array.

1 2 3 4 5 6 7 8Antenna element, n

–2

–1.5

–1

–0.5

0

0.5

Nor

mal

ized

mea

sure

d po

wer

(dB)

0P

/2πPnP~

Figure 4: The normalized power measurement results.

4 International Journal of Antennas and Propagation

investigation is further conducted by 1000 Monte Carlosimulations. The histogram of element amplitude andphase calibration errors is presented in Figures 7 and 8,both of which show nearly normal distributions. More-over, approximately 84% of the amplitude calibrationerrors fall in the region of -1 dB and +1dB, while theprobability percentage of phase calibration error in therange of -5 deg to +5deg is nearly 78%. The standarddeviations of element amplitude and phase estimation

errors for different SNR conditions are presented inFigures 9 and 10. It can be seen from the results thatthe mean square error (MSE) of both amplitude andphase calibration decreases dramatically with the increaseof the SNR. For SNR larger than 30dB (this conditioncan be easily achieved in practice), the amplitude andphased calibration error are less than 0.05 dB and0.5 deg; such high accuracy exceeds most phased arraycalibration demands.

1 2 3 4 5 6 7 8Antenna element, n

–0.6

–0.5

–0.4

–0.3

–0.2

–0.1

0

0.1

0.2

Relat

ive a

mpl

itude

(dB)

Real valueEstimation

Figure 5: The array element relative amplitude and the corresponding estimation results.

1 2 3 4 5 6 7 8Antenna element, n

–20

–15

–10

–5

0

5

10

15

20

Phas

e (de

g)

Real valueEstimation

Figure 6: The array element relative phase and the corresponding estimation results.

5International Journal of Antennas and Propagation

5. Experiment Demonstration

To further validate the effectiveness of the proposed method,a four-element dipole antenna phased array is fabricated andtested. As shown in Figure 11, this antenna array works atS-band with full transmitting and receiving duplex func-tion. For the transmitting mode calibration, a pure sinu-soidal wave source is fed to the phased antenna array,the signal goes to each antenna element from the digitalbeamforming network (DBF) then transmits to the freespace, and a calibration receiver is placed at the far fieldof the phased array and connected with a power meteror spectrum analyzer, while for the receiving mode cali-bration, the calibration transmitter at the far-field region

transmits a pure sinusoidal wave, and the phased arrayreceives the plane wave by the elements and the combinedarray signal is connected to a power meter.

The phased array calibration process is described asfollows:

(1) First, each antenna element is measured individuallyusing a vertical network analyzer (VNA), and theiramplitude and phase distortions are obtained

(2) The combined power of the test array P0 is measuredby a power meter

(3) By setting the DBF weight of the nth antenna elementto zero sequentially, the combined power of the

–2 –1.5 –1 –0.5 0 0.5 1 1.5 2Amplitude estimation error (dB)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

Prob

abili

ty

Figure 7: The amplitude calibration error histogram by 1000 Monte Carlo simulations.

–10 –5 0 5 10Phase estimation error (deg)

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Prob

abili

ty

Figure 8: The phase calibration error histogram by 1000 Monte Carlo simulations.

6 International Journal of Antennas and Propagation

array expecting the nth element Pn is measured bya power meter

(4) By setting the phase of the nth antenna element toπ/2 sequentially, the combined power of the arrayPπ/2 is measured by a power meter

(5) The amplitude and phase calibration results arecalculated by using equation (9) and compared withthe results from step (1)

Both the transmitting and receiving modes are carefullycalibrated in our work. The results of the receiving modeare presented in Table 1, while the results of the transmittingmode are similar and are not given here for the constraint of

20 25 30 35 40SNR (dB)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

MSE

of a

mpl

itude

estim

atio

n (d

B)

Figure 9: The amplitude calibration error under different SNR conditions.

20 25 30 35 40SNR (dB)

0

1

2

3

4

5

MSE

of p

hase

estim

atio

n (d

eg)

Figure 10: The phase calibration error under different SNR conditions.

Dipole antenna element

Integrated T/R module & DBF

Figure 11: The four-element dipole antenna array for experimentdemonstration.

7International Journal of Antennas and Propagation

the paper length. During the experiments, the powermeasurements are accomplished by using Agilent E4405BSpectrum Analyzer, while Keysight M9374A PXIe VectorNetwork Analyzer is used for single-antenna element calibra-tion. The reading accuracy of the power and phase measure-ments is set as 0.1 dB and 0.1 deg, respectively. Figures 12 and13 illustrate the calibration results by our method with theresults from VNA measurement. The results by using Longet al.’s method in [19] are also presented for comparison. Itcan be seen that both the amplitude and phase calibrationresults by our method have good agreement with themeasurement results from VNA and the results fromLong et al.’s method. The corresponding standard devia-tions are 0.06 dB for amplitude calibration and 0.37 degfor phase calibration.

6. Conclusion

Phased array element excitation (both amplitude andphase) always deviates from the ideal values that would

cause array performance degradation, which must be care-fully calibrated and compensated as accurate as possiblefor practical phased array system design. This paper pro-poses a novel power measurement method to obtain theexcitation distortions of individual array elements. Besidesthe total array power, only one phase shift and two powermeasurements for each array element are required to deter-mine the element complex excitation, one by shifting theelement’s phase of π/2 and the other by turning the elementunder test off. This method only involves some simple math-ematical calculation and requires no auxiliary calibration ele-ment or any additional hardware. From the mathematicalpoint of view, it is the minimum required measurements thatuse two scalar values to determine one complex vector. Aneight-element linear patch antenna array is numericallysimulated to demonstrate the calibration procedure, andthe calibration performance is investigated by Monte Carlosimulations. Furthermore, a four-element dipole antennaarray is fabricated and tested to validate the effectiveness ofthe proposed method.

Table 1: Calibration results of the four-element patch antenna array.

Antenna element n = 1 n = 2 n = 3 n = 4

Power measurement results

P0 (dBm) -39.2

Pn~ (dBm) -41.7 -41.7 -41.7 -41.9

Pπ/2 (dBm) -41.6 -41.2 -41.3 -41.3

Calibrationkn (dB) 0 -0.057 -0.057 0.517

xn (deg) 0 6.42 4.78 6.61

VNA resultskn (dB) 0 -0.1 -0.2 0.5

xn (deg) 0 7 4.7 6.3

1 2 3 4Antenna element, n

–0.2

–0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

Relat

ive a

mpl

itude

(dB)

VNA resultsOur methodMethod in [19]

Figure 12: The amplitude calibration results of the four-element patch antenna array.

8 International Journal of Antennas and Propagation

Data Availability

The data used to support the findings of this study areavailable from the corresponding author on reasonablerequest.

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper.

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3

4

5

6

7

Phas

e (de

g)

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Figure 13: The phase calibration results of the four-element patch antenna array.

9International Journal of Antennas and Propagation

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