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Summer Research Fellowship Programme of India's Science Academies

Study and analysis of fault tolerances in a radio interferometer

Varsha Bharti

Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103

Prof. R. Ramesh

Indian Institute of Astrophysics, Gauribidanur 561208

Abstract

Radio astronomical observations require high angular resolution and sensitive beams to observe distant objects with angular dimensions of a few arc minutes or less, for example the Sun. For a telescope with such a high resolution, its aperture size should be large enough, but it is practically very difficult to establish a single antenna setup with very large aperture. Through interferometry, multiple antennas can be brought together to achieve resolution comparable to one very large antenna system. Moreover with an array, antenna beams with high directivity and sensitivity can be achieved. Radio interferometry most commonly involves antennas or antenna arrays of identical design. For a large number of discrete antennas, the far-field pattern is similar to the pattern of a continuous array of same length and its fourier transform can give us aperture field distribution due to reciprocity. Our primary interest is to study solar emissions at low frequency, i.e. a few MHz. For this, we have simulated an antenna array of two groups with eight identical antennas each combined as an adding interferometer. The two groups are behaving as two elements of a multiplying interferometer. We are observing both in-phase and out-phase components of multiplying interferometer with different base length configurations and compared it with real-time data obtained from Gauribidanur Radio Heliograph. As base length increases, the resolution improves and beam becomes more directive/sensitive. Since many antennas are involved in receiving signals from the source, due to practical conditions some antennas/ antenna signal path may get faulty or some unwanted phase may be introduced which leads to disturbed electric field pattern and power pattern. The objective of this work is to study and quantify a minimum tolerance level of faults (amplitude and phase variations) up to which overall power pattern of the interferometer does not get affected significantly through pattern analysis and Fourier transform analysis.

Keywords: Gauribidanur Radio Heliograph, Fourier transform analysis, antenna, sensitivity, resolution, beamwidth.

Abbreviations

Abbreviations
GRH/GRAPH Gauribidanur Radio Heliograph
IAS Indian Academy of Science
LPDA Log-Period Dipole Antenna
FFT Fast Fourier Transform
HPBW Half Power beamwidth
NPBW Null Power beamwidth

INTRODUCTION

Background

The Gauribidanur RAdio heliograPH (GRH ; Ramesh et al. 1998) is a two-dimensional T-shaped radio interferometric array with two orthogonal arms of well-defined distribution of antenna elements extending along the east-west (EW) and south (S) direction. The basic receiving element of the array is a log-periodic dipole (LPDA) which is capable of receiving a wide band of signal with a constant gain. There are a total of 384 LPDs configured as 64 similar groups. Each group in the EW direction consists eight antenna elements with an inter-element spacing of ≈ 10m while each group in the NS direction consists of four antenna elements with an inter-element spacing of ≈ 7 m. The EW arm of the GRAPH consist of 32 similar groups that extends to a length of ≈ 2560 m. The spacing between adjacent group centres is 80m except between group 16 and 17 which is 90m due to T-intersection. It is used to take two dimensional images of solar corona at low frequencies in range 40 MHz to 150 MHz No other radio telescopes, dedicated for solar observations, are presently operational in the above frequency range anywhere in the world. The basic receiving element is a log period dipole antenna (LPDA) which can observe wide band of frequencies. Since, the spacing between each element is greater than wavelength of observation, grating lobes can be seen in the field pattern.

Untitled Diagram.jpeg.jpg
    E-W array arrangement of GRAPH

    Statement of the Problems

    The GRAPH array is in regular operation throughout the day and it is extended over nearly 5 kilometer. Thus, due to electronic failure, environmental conditions or system irregularities, the array elements are prone to failure in practical scenario. However, it is difficult to determine the number of faulty elements and their position within the array presently. The antenna failure introduces error in the overall received pattern of the array which may lead to several other complications while imaging the solar corona. In many situations, we can live upto a certain distortion in the overall field pattern because it is not easy to replace the faulty elements easily.

    Objectives of the Research

    • To study and quantify a minimum tolerance level of faults (amplitude variations) up to which overall power pattern of the interferometer does not get affected significantly through pattern analysis and Fourier transform analysis and also identify faulty groups in the array.
    • To identify number of faulty elements within a group from fringe pattern using fast Fourier transform (FFT). and to study their effect on over all radiation pattern of the interferometer.

    LITERATURE REVIEW

    Radio Telescopes and Interferometry[3]

    Radio waves can penetrate the Earth’s atmosphere whose wavelength ranges from few millimeters to nearly 100 meters. Radio astronomy deals with the study of the universe using radio frequency electromagnetic signals that are emitted from different objects in the universe and can be detected on or near Earth using radio telescopes. Radio telescopes can be used to study continuum emission from radio sources or its time variability. To detect waves coming from far sighted objects, the resolution of the instrument should be very high. Resolution means how two closely spaced points in the sky can be distinguished as two separate objects. For a telescope, resolution is given by ≈λ/D. Radio waves have large wavelengths. Thus, to detect radio wavelengths a very large aperture telescope is required which practically not feasible. Instead interferometers are used whose resolution is similar to that of a continuous aperture using multiple antennas of smaller apertures. The distance between two or more telescopes in the array can be large to improve the resolution and sensitivity of the instrument.

    pic.png
       An interferometer as a telescope with larger aperture 

      Antenna Array and Adding Interferometer

      An antenna array is used to obtain a directed beam to obtain signals from a desired source. The number, geometry and phases of array elements depends on desired angular pattern. Consider a linear array of N isotropic points uniformly distributed with spacing d is shown in fig.3. It is assumed that beam is formed at the centre of the array. Then, the far-field electric field pattern of the array is given by :

      Untitled Diagram.jpg
        Eight element antenna array with beam at centre

        E=i=1NEi.((N+1)/2i)jΨ)E = \sum_{i=1}^N{E_i.^{-((N+1)/2-i)j\Psi)}}

        where, Ψ  =  2πλdsin(θ)\Psi\;=\;\frac{2\pi}\lambda d\sin\left(\theta\right)

        Here, Ei is the electric field pattern of ith antenna element, λ is the wavelength of and θ is the angle at which source is present.

        For a uniform array, electric field response of all antenna will be equal, say E0 , then the total field-pattern will be,

        E  =E0  sin(Nψ/2)sin(ψ/2)\begin{array}{l}E\;=E_0\;\frac{\sin(N\psi/2)}{\sin(\psi/2)}\\\end{array}

        here, E0 represents individual element and sin(Nψ/2)/sin(ψ/2) represents the array factor. The far-field pattern of a single element is generally taken to be a gaussian function of suitable width.

        The parameters that affect the form of the radiation pattern of a linear array are the spacing between elements, the amplitude and phase excitation of each element and the radiation pattern of a single element. These parameters can be set in a suitable way in order to obtain the desired radiation pattern

        As the number of elements in array increases, the aperture can be assumed to be continuous.

        The fourier transform of the aperture-distribution in spatial frequency domain gives electric far-field radiation pattern and vice versa. Thus, by taking fourier transform of the far-field pattern, the aperture field distribution can be determined. Also, the fourier transform of the antenna power pattern is proportional to complex auto-correlation of the aperture distribution.

        eightelement1_1.jpg
          Patterns of an eight element array

          Multiplying Interferometer

          In a two-element multiplying interferometer, the patterns of the two elements are correlated(multiplied and averaged). A simple two-element multiplying interferometer is shown in fig. 2.

          correlation3_2.gif
            A 2-element multiplying interferometer

            The multiplying interferometer output with average value zero. The power pattern variation in both adding and multiplying interferometer have been shown. The adding interferometer behaves similar to a low pass filter whereas an multiplying interferometer behaves as a band pass filter.

            adding vs multiplying2.jpg
              Power Pattern of adding interferometer vs Power Pattern of multiplying ineterferometer

              If an adding interferometer with sufficient number of elements is assumed to be a group and two such similar groups are assumed to be elements of a multiplying interferometer separated by some baselength, fringes are obtained. The fourier transform of these fringes will give peaks at their respective spatial frequency.

              fringe.1jpg_1.jpg
                Fringe pattern at frequency 53.3 MHz for a 2-element multiplying interferometer with baselength 2490m and whose each element is an array of 8-elements separated by 10m.

                METHODOLOGY

                The pattern obtained by each group is a result of superposition of waves radiated by each element in the array at the source. Thus absence/fault in one of the elements will result in absence of that wave component from the overall field pattern leading to variations in main lobe and side-lobe intensities. Hence, if all the elements have same phase the maximum amplitude of the overall radiation will depend on the number of non-faulty elements.

                In order to detect the faulty group one group of the array, say group 1, is assumed to be the test group whose all elements are in perfectly working condition. The fringe pattern obtained by multiplication of group 1 with all other group is considered for analysis. The fast Fourier transform of the fringe pattern will give a peak at spatial frequency of the respective fringe i.e. at base length per unit wavelength for the specific configuration. The amplitude of the fast-fourier transform peak is proportional to the number of non-faulty elements in the group analysis of which will give the number of faulty elements in the group.

                Joining the peaks obtained in the spatial frequencies for all 32 baselengths together is proportional to the aperture of the over-all system which is equivalent to a high resolution telescope and thus in time domain, it will be proportional the intensity of the radiating field of the system. Hence if there are faults in the array elements, the aperture distribution will not be uniform and hence there will be distortion the radiating field.

                An interferometer with two groups each having an array of 8 elements has been simulated which is similar to the geometry of E-W array of GRAPH. Total 31 different base length configurations are simulated and analysed with group 1 being common to all the combinations in both time and spatial frequency domain. The simulated results are then implemented on real time data obtained from GRAPH for calliberator Cygnus for two frequencies viz. 53.3 MHz and 80 MHz.

                RESULTS AND DISCUSSION

                Interferometer Response with faults

                The amplitude of the fringe is proportional to the product of the number of non-faulty elements in each group. Say, m elements are non-faulty in group 1 and n elements are non-faulty in group 2, then the overall fringe amplitude will be proportional to (m×n). This variation is independent of base length configuration as well as frequency and can be interpreted from the MATLAB® simulations shown.

                fault_80m1.jpg
                  Change in fringe amplitude with number of faulty elements for 80m baselength at 53.3 MHz
                  fault24901_1.jpg
                    Change in fringe amplitude with number of faulty elements for 2490m baselength at 53.3 MHz

                    In the spatial frequency domain, each base length will have corresponding peak at base length per unit operating wavelength distance from center. The amplitude of this peak is also proportional to the number of operating elements in the group. Thus, by taking group 1 to be perfect, number of faulty elements in the other groups can be determined by quantising the amplitudes.

                    allokay.jpg
                      Quantised Spatial frequency components of all 31 baselength with no faults at 53.3 MHz
                      faultytransform1_1.jpg
                        Spatial frequency response of faulty array at 53.3 MHz. Here, group 2 has 1 faulty element, group 4 has 2 faulty element, group 15 has all elements faulty and group 30 has 3 faulty elements. The red circle indicates fault cases while black asterisks represent ideal scenario.
                        word.jpg
                          Faulty group identification at 53.3 MHz. Here, group 2 has 1 faulty element, group 4 has 2 faulty element, group 15 has all elements faulty and group 30 has 3 faulty elements. The red circle indicates fault cases while black asterisks represent ideal scenario.
                          spat80.jpg
                            Spatial frequency response of faulty array at 80 MHz. Here, group 3 has 5 faulty element, group 7 has 3 faulty element, group 9 has 4 faulty elements, group 16 has 7 elements faulty, group 22 has 5 faulty elements and group 32 has 3 faulty elements. The red circle indicates fault cases while black asterisks represent ideal scenario.
                            fault80.jpg
                              Faulty group identification at 80 MHz. Here, group 3 has 5 faulty element, group 7 has 3 faulty element, group 9 has 4 faulty elements, group 16 has 7 elements faulty, group 22 has 5 faulty elements and group 32 has 3 faulty elements. The red circle indicates fault cases while black asterisks represent ideal scenario.
                              • Overall radiating field pattern in ideal case

                              From the spatial domain plots shown above, the following aperture distribution for the interferometer is obtained upon joining the peaks when there is no fault in the system and from fourier transform ananlysis the corresponding far-field pattern is obtained. A stepwise approach to generate these patterns is explained in section 6.1.

                              cygnus_ideal.jpg
                                Ideal aperture distribution and corresponding far-field pattern
                                • Radiating field pattern comparison for data obtained from GRAPH for CYGNUS calliberator

                                The aperture distribution and its corresponding far-field pattern for a calliberator sourec Cygnus is shown for data obtained from GRAPH as compared to the ideal case. It can be seen that the pattern is getting distorted due to irregularity in system in the practical scenario. Thus, by studying its effect system can be further improved for better reception of signals from the astronomical sources.

                                cygnus_2.jpg
                                   Aperture distribution and corresponding far-field pattern of Cygnus 

                                  CONCLUSION

                                  The amplitude of fringes obtained from interferometer and its spatial frequency response depend upon the number of working antennas in the system. A decrease in amplitude of peak of corresponding spatial frequency component in the overall spatial frequency response implies fault in the correspondng group of the sytem. Also, if there is any error in the testing group all the groups will be affected equally. As the number of faulty antenna in a group increases, the fringe amplitude decreases as well as the peak amplitude its spatial frequency component decreases. By measuring the decrease in fringe amplitudes, an estimate can be given about the number of faulty antennas in the array. Further analysing the fringe pattern of the faulty group, can give us a better estimation of the number of faulty elements in the array. It is found that the fringe amplitude reduction due to faulty elements is independent of the base length configuration. The overall spatial doamin response is continuous and uniform in ideal case and thus if all the antennas are assumed to be equally contributing, the overall far-field pattern will be sinc function of very narrow HPBW and hence capable of detecting small sources with sharp intensity. In the practical case, however the spatial domain response is distorted due to sytem limitations. Thus, the obtained far-field pattern is distorted from ideal case. This distortion is tolerable unless signal recieved is not hampered significantly.

                                  This work can be further extended by developing algorithms to detect the locations of faulty group within the array by employing newly evolving artificial neural network techniques, so as to compensate the error and improve the performance of the system. Further, the overall radiation patterns can be analysed to obtain a statistical deviation from expected response due to the error introduced. in the system, thereby enhancing the system capabilities.

                                  REFERENCES

                                  1. Bracewell Ronald N. (2000), The Fourier Transforms and its applications.Singapore : McGraw-Hill Higher Education.

                                  2. James J. Condon and Scott M. Ransom (2016), Essential Radio Astronomy. Princeton, NJ: Princeton University Press.

                                  3. Kraus John D. (1986), Radio Astronomy. New Hampshire : University of New Hampshire Printing Services.

                                  4. Ramesh, R., Subramanian, K. R., Sundara Rajan, M. S. & Sastry, Ch. V.(1998). Solar Phys., 181, 439.

                                  5. Superposition of waves. Retrieved from http://www.physics.okstate.edu/ackerson/Physics3213/Superposition_2.html​

                                  ACKNOWLEDGEMENTS

                                  As a student, I always seek for good learning experience and my stay at Gauribidanur gave me numerous scientific as well as life lessons. It is my pleasure to be indebted to various people, who directly or indirectly contributed in the development of this work and who influenced my thinking, behavior, and acts during this internship. I express my sincere gratitude to Science Academies' SRF Programme, for providing me an opportunity to undergo this summer internship under the guidance of Prof. R. Ramesh at Indian Institute of Astrophysics, Gauribidanur. This work would not have been possible without his constant support and encouragement.

                                  I am thankful to Dr. C. Kathiravan, Mr. G.V.S. Gireesh and Mr. Indrajit Barve for their support, cooperation and motivation during the internship. I would also like to thank all the observers for providing data for my work. I also extend my sincere appreciation to Mr. A. Anwar and everyone at Gauribidanur Radio Observatory for their hospitality and affection.

                                  Lastly, I would like to thank the almighty and my parents for their moral support and also my friends and fellow beings at GRO with whom I shared my day-to-day experience and have helped me to make my stay at this observatory peaceful.

                                  APPENDICES

                                  1. Stepwise approach to generate far-field pattern of overall system

                                  Step 1: Load data of corresponding source

                                  Step 2: Take fourier transform for fringes of each baselength and plot them together in the spatial frequency domain. You can now visualise a continuous aperture distribution.

                                  Step 3: Join the peaks of the pattern obtained in Step2. It will be proportional to the aperture distribution of the interferometer.

                                  Step 4: Take fourier transform of the pattern obtained in Step 3. This will give a profile depicting the over all radiation pattern of the interferometer.

                                  Step 5: This pattern can be compared with the expected/ideal case for analysis of the system performance.

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