Complex gain modeling of CHIME’s coaxial cablesbySidhant GulianiA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORAL STUDIES(Physics)The University of British Columbia(Vancouver)August 2018c© Sidhant Guliani, 2018Committee PageThe following individuals certify that they have read, and recommend to the Faculty of Graduate and Post-doctoral Studies for acceptance, the thesis entitled:Complex gain modeling of CHIME’s coaxial cablessubmitted by Sidhant Guliani in partial fulfillment of the requirements for the degree of Master of Sciencein Physics .Examining Committee:Mark Halpern, PhysicsSupervisorGary Hinshaw, PhysicsSupervisory Committee MemberAbstractCHIME is a new radio interferometer located at the Dominion Radio Astrophysical Observatory (DRAO) inPenticton, BC. The primary goal of CHIME is to constrain the dark energy equation of state by measuringthe expansion history of the Universe using the Baryon Acoustic Oscillation (BAO) scale as a standardruler. CHIME consists of 4 cylindrical reflectors, each populated with 256 dual-polarization antennas alongits focal-line. Prior to digitization, each signal chain consists of a low noise amplifier, 50m of coaxial cable,and a filter amplifier. In order to obtain accurate interferometric imaging, we need to determine the relativecomplex gain (amplitude and phase vs. frequency) of each analog chain to 0.3%. The complex gain of eachreceiver depends primarily on temperature. This thesis discusses efforts to construct a thermal model of theCHIME’s coaxial cables that will allow us to meet our calibration requirements.iiiLay SummaryThe Canadian Hydrogen Intensity Mapping Experiment (CHIME) is a transit interferometer at the DominionRadio Astrophysical Observatory (DRAO) in Penticton,BC, Canada. CHIME will map neutral hydrogen inthe frequency range 400 – 800 MHz over the northern sky, producing a measurement of baryon acousticoscillations (BAO) at redshifts between 0.8 – 2.5 to probe dark energy. In this thesis we are going to discussabout the efforts to construct the thermal model for the CHIME’s coaxial cables.ivPrefaceA discussion among myself and Mark Halpern led to design of my research program. The experimentaldesign was proposed by me under the supervision of Mark Halpern and was deployed on-site with the helpof Meiling Deng, Mandana Amiri and Mark Halpern. The analysis for the project was done by me underthe guidance of Mark Halpern and Gary Hinshaw.vContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vContents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiAcknowledgement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Where does dark energy came from? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Cosmological model for DE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 BAO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 How can we see BAO in 21cm Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 CHIME overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 Calibration requirements of CHIME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1 Why is Calibration required . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 21cm Signal in CHIME data set and foreground removal technique . . . . . . . . . . . . . . 92.3 Calibration requirements for CHIME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.4 Thermal model for components in analog chain . . . . . . . . . . . . . . . . . . . . . . . . 143 Complex Gain measurement of cables: Properties & Experimental setup . . . . . . . . . . . 183.1 Properties of LMR-400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2 Delay measurement Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2.1 Swarup and Yang system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.2 Frequency Offset round-trip scheme . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Experimental setup for complex gain measurements. . . . . . . . . . . . . . . . . . . . . . 223.4 Geometrical layout of the cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23vi4 Delay Measurement & Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1 Delay Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Delay Modeling & Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.3 Non-obvious grouping for delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 Attenuation measurement & modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.1 Attenuation in Cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.2 Reflections in cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.3 Thermal Model for Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.4 Residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55viiList of FiguresFigure 1.1 History of Universe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Figure 1.2 Baryonic Acoustic oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5Figure 1.3 Hyperfine splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6Figure 1.4 CHIME analog chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7Figure 2.1 Mode Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12Figure 2.2 Example of foreground filtering from simulated sky. . . . . . . . . . . . . . . . . . . . 13Figure 2.3 Biasing of power spectrum for complex gain perturbations. . . . . . . . . . . . . . . . . 14Figure 2.4 Biasing of power spectrum for beam perturbations. . . . . . . . . . . . . . . . . . . . . 15Figure 2.5 Thermal model for Low noise Amplifiers(LNA) . . . . . . . . . . . . . . . . . . . . . . 16Figure 2.6 Thermal model for Filter-Amplifiers(FLA) . . . . . . . . . . . . . . . . . . . . . . . . 16Figure 2.7 Crude thermal model for cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 2.8 Complex gain when we initialize the thermal chain with LNA, cables and amplifiers alltogether. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Figure 3.1 Phase spectra of cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Figure 3.2 Residual of phase spectra after linear fit . . . . . . . . . . . . . . . . . . . . . . . . . . 20Figure 3.3 Residual of phase spectra after polynomial fit . . . . . . . . . . . . . . . . . . . . . . . 21Figure 3.4 Swarup and Yang experimental setup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 3.5 Frequency-Offset round-trip scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 3.6 Round trip phase measurement scheme experimental setup . . . . . . . . . . . . . . . . 24Figure 3.7 Setup of Noise source and correlator inside the hut. . . . . . . . . . . . . . . . . . . . . 25Figure 3.8 Setup of 8 channel correlator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 3.9 Geometrical layout of cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26Figure 4.1 Measured delay of the cables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 4.2 Scaled delay of the cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Figure 4.3 Residual of scaled delay and scaled air temperature. . . . . . . . . . . . . . . . . . . . . 30Figure 4.4 Residual when air temperature is taken as reference. . . . . . . . . . . . . . . . . . . . 31Figure 4.5 Residual when delay of cable A00 is taken as reference. . . . . . . . . . . . . . . . . . . 32Figure 4.6 Non-obvious grouping 1 in delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33viiiFigure 4.7 Non-obvious grouping 2 in delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Figure 5.1 Measured gain of the cables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Figure 5.2 Gaing spectra of a single round trip path at an instant and linear fit to the gain spectra . . 36Figure 5.3 Standing wave spectra of seven round trip paths at an instant. . . . . . . . . . . . . . . . 37Figure 5.4 Standing wave spectra of a single cable for 500 time frames. . . . . . . . . . . . . . . . 38Figure 5.5 Standing wave spectra of a coldest and the hottest time . . . . . . . . . . . . . . . . . . 38Figure 5.6 Scaled reflection spectra for a single cable and 500 time frames. . . . . . . . . . . . . . 39Figure 5.7 Scaled deviations as function of time for the reflections in a single cable. . . . . . . . . 40Figure 5.8 Delay spectra to see the reflection terms . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 5.9 Spectra of attenuations due to reflections in short cables. . . . . . . . . . . . . . . . . . 42Figure 5.10 Spectra of attenuation due to reflections in long cables. . . . . . . . . . . . . . . . . . . 43Figure 5.11 Scaled deviations for the attenuation due to reflections in short cables . . . . . . . . . . 44Figure 5.12 Scaled deviations for the attenuation due to reflections in long cables . . . . . . . . . . . 44Figure 5.13 Slope of linear fit as function of time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45Figure 5.14 Scaled Y-intercept of linear fit as function of time for 7 channels. . . . . . . . . . . . . . 45Figure 5.15 Residual for for each component in attenuation at 600MHz . . . . . . . . . . . . . . . . 46Figure 5.16 Residual for gain at 600MHz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 5.17 Algorithm for attenuation modeling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 6.1 Amplitude fluctuations and linear fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Figure 6.2 Thermal coefficient of the amplitude fluctuations as a function of frequency for channel53. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Figure 6.3 Eigenvalues for 2 frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53Figure 6.4 Thermal coefficient of the amplitude fluctuations as a function of frequency for channel10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54ixAcknowledgementI would like to thank Mark Halpern, my supervisor, for his support, advice, encouragement and amazingproblem solving. I would like to thank Gary Hinshaw for his patience, encouragement and guidance. Iwould also like to thank Mateus Fandino, Meiling Deng, Richard Shaw, Mandana Amiri and Don Wiebe fortheir help, guidance and patience in the lab. I am thankful for all the support I have received from the wholeCHIME team. I would like to thanks my sister and parents for their support and encouragement during thisjourney.xChapter 1IntroductionThis section is about the introduction to Dark energy and an experiment, Canadian Hydrogen IntensityMapping Experiment (CHIME) currently under commissioning at Penticton, BC, Canada. This radio tele-scope is primarily intended to probe the nature of dark energy. CHIME will measure the distribution ofneutral hydrogen in the universe by observing the 21cm hyperfine transmission line in the redshift range0.8 < z< 2.5.1.1 Where does dark energy came from?Figure 1.1 describes the evolution of universe from Big Bang to present. The estimate of the earlier stagesof universe existence is around 13.8 billion years ago. Early in the history of the universe, there was a periodcalled inflation period where there was exponential growth of universe size.After inflation, universe was left with protons, electrons, protons, neutrons, neutrinos, dark matter andother form of energies. Photons interacted with these charged particles via Thompson scattering and weretrapped in the hot dense plasma. As the time goes on, universe started to expand and cooling down. Whenthe universe temperature reached 3˜000K, electrons and protons started to interact to form neutral hydrogen.This period in the history of universe is called recombination. Recombination happened when the universewas 4˜00,000 years old, at the redshift (z) = 1089. During this period, the universe became transparent tophotons, as the photons interact to much lesser degree with neutral matter. This process made photons todecouple from matter and travel the universe. The photons which emitted after the recombination, are theCosmic Microwave Background (CMB). We are seeing these photons every day at every direction at 2.7K.Eventually, the neutral hydrogen condenses due to the attractive gravitational force. The core of theselocal neutral hydrogen has higher temperature and pressure, which results in the nuclear reaction in thecore. This is how the stars were formed. These local stars formed the first galaxy. During this process,it is expected the expansion of universe to decelerate due to the attractive gravitational force between thegalaxies. But, during this process, the expansion of universe started to accelerate. A direct measurement tosupport this acceleration comes from supernovae observation by measuring their luminosity distances. Oneof the model to explain this acceleration is dark energy.1Dark energy is the name given to the antigravity-like agent that accelerates the expansion of universe[3], [4]. Based on the Cosmic Microwave Background (CMB) measurements, the total mass-energy of theuniverse is comprised of 4.6% baryonic matter, 22.7% dark matter and 72.8% dark energy. Understandingdark energy is one of the biggest challenge of cosmology today, [5]. The dark energy can be explainedby cosmological constant Λ, which prescribes a constant energy density everywhere. The cosmologicalconstant has negative pressure equal to its energy density and so causes the expansion of the universe toaccelerate. In the next section we will be discussing the cosmological model for dark energy.1.2 Cosmological model for DEThe Equation of state is characterized by a dimensionless number w(a), equal to the ratio of its pressure Pto its energy density ρ :w(a)=Pρ= wo+(1−a)wa (1.1)where a is the scale factor, related to redshift by a=1/(1+z); wo is the present value of w, and wa param-eterizes the evolution of w(a).On the large scale, universe is isotropic and homogeneous. By isotropic it means that the universe issame in all directions and by homogeneous, it means that the universe is not the function of position, [6].Friedman-Robertson-Walker metric is the equation that describes the expanding universe. The matric isgiven by:ds2 = -dt2+ a2(t)[dr21− kr2 + r2(dθ 2+ sin2θdφ 2)] (1.2)here k is a constant representing the curvature of space (K>0: closed universe, k<0: Open universe,and k=0: flat universe). Using the Friedman metric and Einstein field equations of general relativity gives,Friedman equations:a¨a=−4piG3(ρ+3P)+Λ3(1.3)(a¨a)2 = H2(t) =8piGρ3− ka2+Λ3(1.4)here, H(t) is the hubble parameter; vacuum energy(dark energy) is defined as ρΛ =Λ/(8piG) = constant.From the last two equations,ρ˙ =−3H(ρ+P) (1.5)In the next step we are assuming that all the components of energy density are perfect fluids that follows theequation of statePi = wiρi (1.6)2Figure 1.1: A figure to illustrate the evolution history of the universe over13.7 billion years. Timeevolves in the horizontal direction. 3D universe at a particular time is depicted by a cross sectionof the grid. Image credit:NASA/WMAP Science Team.ρi = a−3(1+wi) (1.7)where w = 0,1/3,-1 for radiation, matter and cosmological constant. In the next step we are goingto introduce the critical density. Critical density is the parameter which defines the overall geometry ofuniverse. In λCDM model there are three important density parameters due to baryons, dark matter anddark energy. The spatial geometry of the universe as measured by WMAP is nearly flat. This means that theuniverse can be well approximated by a model where the spatial curvature parameter k is zero.ρc =3H28piG(1.8)The first Friedman equation is given by:=3H2H20=Ωra−4+Ωma−3+Ωka−2+ΩΛ (1.9)Ωk is the spatial curvature density, Ωk < 0 for closed universe, Ωk > 0 for open universe and Ωk = 0 forflat universe. If the the equation of state for dark energy has w 6=−1 or if the w changes with the sale factor,then the Friedman equation can be generalized with:Ωk→ΩΛexp{3∫ 1ada′a′[1+w(a′)]}. (1.10)1.3 BAOThe early universe consisted of hot and dense plasma of photons and baryons, tightly coupled via ThompsonScattering. This region of over-density attracts matter due to gravity. The heat produced by the interaction ofphoton-matter creates outward pressure. These opposing forces of gravity and pressure created oscillations,analogous to sound waves created in air by pressure differences. These oscillations are seen in cosmicmicrowave background anisotropy data and they also leave a hint in the clustering of galaxies and mattertoday. These oscillations results in an outgoing spherical wave from the over dense spot. The photonsand charge particles travel together till the recombination period, while the dark matter stays at the overdense spot as they interact only gravitationally. These spherical waves travel till recombination. Neutralhydrogen stays in the spherical shell and keep attracting local matter to the shell due to gravitation. Thedistance between the over-dense spot and the spherical shell is referred to as sound horizon. Its size changesproportional to size of universe. Many spherical shells are created, analogous to the pebble dropped in thepond.The BAO can be detected using the two point function across the sky. BAO are referred to as the standardruler as the comparison of its angular size with respect to redshift can tell about the expansion history ofuniverse, given that its co-moving size is known to be 150 Mpc.1.4 How can we see BAO in 21cm RadiationMapping the distribution of neutral hydrogen using the 21 cm hyperfine spin flip transition is a promisingmethod of measuring the evolution of the BAO standard ruler. BAO can be seen in neutral hydrogen distri-bution as neutral hydrogen traces the matter distribution on large scale. Neutral hydrogen emits/absorbs theradiation at 1420 MHz due to hyperfine splitting. This wave is 21cm radiation, which we are trying to map.In the redshift region, ranging from z=0.8-2.5, the apparent frequency of 21cm signal(neutral hydrogen) isshifted to 400-800 MHz. Therefore, measuring the 21cm signal as function of frequency we are measuringthe matter distribution as function of redshift.4Figure 1.2: When the inflation began, the baryon gas and photons traveled together, away from darkmatter. At z = 1440, recombination occur, photons are no longer coupled to matter by Comptonscattering and they travel freely in universe. Later, baryons and dark matter collapse towardseach other but density enhancement at the sound horizon remains, which we can see at the lowerright plot, a bump at 150 Mpc. Refer to [10].5Figure 1.3: Hyperfine splitting of the neutral hydrogen resulting in emission of 21cm radiation.This radiation comes from the transition between the two levels of hydrogen 1s ground state,slightly split by interaction between electron spin and nuclear spin. This splitting is knownas hyperfine splitting. The state in which the electron and neutron have the parallel spins,have slightly more energy compared to the state when their spins are anti-parallel. [Credits:www.britannica.com/science/hyperfine-structure]1.5 CHIME overviewCHIME (Canadian Hydrogen Intensity Mapping Experiment) is a radio interferometer at the DominionRadio Astrophysical Observatory (DRAO). The CHIME consists of four adjacent cylindrical dishes, each20 m wide by 100 m long and featuring an array of close-packed feeds along the focal lines. The stationarycylinders are oriented north-south to form a transit telescope as the sky rotates from east to west.Each cylinder is equipped with 256 dual- polarization antenna feeds sensitive to 400-800 MHz. Thereare total of 1024 feeds and 2048 low-noise amplifiers (LNAs) and are located along the focal line. Theanalog chain of the CHIME consists of the components shown in Figure 1.4. Signals are received fromsky through feeds on the focal line and are amplified by LNA. These signals are transferred over 50mcoaxial cables to a shielded RF room. Then the signals are bandpass filtered, amplified using FLA (FilterAmplifiers). These amplifiers are in the RF room, which is thermally controlled. These analog signalsare digitized by ACD’s (Analog to Digital Converter) and then Fourier transformed and correlated by FXcorrelator for data analysis. The receiver huts are air and water cooled.6Figure 1.4: The analog system signal chain overview. The feeds receives signals from sky. Thereceived signals are amplified by the low noise amplifiers by 42dB across the 400-800 MHz. Thesignals are then transmitted over 50m of LMR-400 coaxial cables to RF shielded room. Thesignal is then bandpass filtered to 400-800 MHz and is further amplified by 34 dB, The powerreceived at the input of ADC is -19 dBm. [Credit: Plots by Mandana Amiri and Rick Smiegel]CHIME will map the large-scale distribution of neutral hydrogen gas in the universe by directly detectingits red shifted 21 cm radiation. By measuring the scale of the Baryon Acoustic Oscillations (BAO) acrossthe redshift range z=0.8-2.5, CHIME will study the epoch when Dark Energy generated the transition fromdecelerated to accelerated expansion of the universe and constrain the properties of the Dark Energy withunprecedented precision.7Chapter 2Calibration requirements of CHIMEIn this section we are going to discuss about the m-mode formalism, which is a method to analyze thetransit interferometers. Followed by an overview of Karhynen-Loeve scheme for foreground calibrationrequirement.2.1 Why is Calibration requiredCHIME need to produce sensitive and foreground-cleaned maps. To measure BAO from power spectrum,we need to eliminate the systematic errors due to foreground and the instrumental gain. The main driver forcalibration requirements is the brightness of the galactic signal. For example, frequency-dependent phaseerrors will place a fraction of bright galactic emission in the wrong direction, resulting in systematic errorsin the sky maps that may corrupt the BAO signal. The calibration requirements for CHIME are set usingend-to-end simulations of the CHIME experiment. Based on these simulations we need to know the complexgain of each receiver to 0.3% within each minute in order not to bias the derived power spectrum by morethan its statistical uncertainty, discussed in more detail in section 2.3.The challenge for this experiment is to separate the 21cm signal from astronomical containments whichare 103−105 times brighter than the 21 cm signal. This emission comes mainly from synchrotron radiation,which is spectrally smooth, and in principle this allows it to be separated from the 21 cm as it is describedby a small number of modes (components) and can be easily removed. The remaining components whichcontains the significant spectral structure are assumed to be free from contamination. There are some affectsthat complicate foreground removal:1. Mode Mixing: Mode-mixing describes the mixing of angular structure to frequency structure as thebeams are frequency dependent. This phenomena is discussed in more detail in section 2.2.2. We don’t have proper understanding of the small angular and frequency scales of astrophysicalforegrounds.In section 2.2, we will develop an intuition for the spherical harmonic transient telescope formalism andforeground filtering technique as described in [2] and in section 2.3, we will be discussing the calibrationrequirement of CHIME as described in [2].82.2 21cm Signal in CHIME data set and foreground removal techniqueMapping the universe with 21cm line requires rapidly probing large volume of data using wide field ratiotelescopes. An transient telescope can be seen as collection of feeds, fixed relative to ground. Each feed(Fi)measures the combination of electric field coming from different parts of sky. In interferometry, visibilityVi j is the instantaneous correlation between two feeds, Fi and Fj.Vi j =< FiFj >=∫d2nˆAi(nˆ)A∗j(nˆ)e2pi nˆ.ui jT (nˆ) (2.1)here ui j = (ri− r j)/λ is the spatial separation between the two feeds, divided by wavelength nˆ is theposition where feed is pointing to in the celestial sphere and Ai(nˆ) is the beam of feed i.Ωi j =√ΩiΩ j (2.2)Ωi =∫d2nˆ|Ai(nˆ)|2 (2.3)Ωi j is the geometric mean of the individual beam solid angles. The visibilities are normalized so thatthey are temperature like. The azimuthal angle(φ ) dependence of the visibility came from the rotation ofearth. The visibilities are also corrupted by the instrumental noise which we take into account by addingni j(φ) in visibility equation. Rewriting the visibility equation in terms of transfer function, Bi j:Vi j(φ) =∫dΩBi j(nˆ,φ)T (nˆ)+ni j(φ) (2.4)where transfer function is defined as,Bi j(nˆ,φ) =1Ωi jAi(nˆ,φ)A∗j(nˆ,φ)e2piinˆ.ui j(φ) (2.5)As Vi j is periodic, it can be decomposed in its Fourier series and is given by:vi jm =∫ dφ2piVi j(φ)e−imφ (2.6)In next step we want to expand the beam transfer function (Bi j) and sky brightness temperature (T)mentioned in equation 2.4 in terms of spherical harmonic functions (alm),T (nˆ) = ΣalmYlm(nˆ), (2.7)Bi j(nˆ,φ) = ΣBi jlm(φ)Ylm(nˆ) (2.8)the reason why we are expanding the beam transfer function in terms of spherical harmonics, will bebecome apparent very soon.9vi jm =∫ dφ2pi{∫dΩ[ΣalmBi jlme−im1φYlmYl1m1]}e−imφ (2.9)=∫ dφ2pi{ΣalmBi jlmeim1φ[∫dΩYlmYl1m1]}e−imφ (2.10)=∫ dφ2pi[ΣalmBi jlmeim1φ]e−ikφ (2.11)= ΣalmBi jlm[∫ dφ2pie−i(m1−m)φ](2.12)vi jm = ΣalmBi jlm+ni j(φ) (2.13)Equation 2.13 describes the way sky information contained in alm maps into the observed data (vi j) giventelescope design (Bi j and ni j). For a given m and frequency v, the measured visibilities are the projection ofl-modes on the sky for the measured m. For ease of notation, we will introduce a label α which indexes boththe positive and negative m parts of all included feed pairs i j, such that any particular α specifies exactlythe values of i j, ±, So that we can write Equation 2.13 as:vαm = ΣalmBαlm+nαm (2.14)In the next step we will write transfer function, visibility and harmonics coefficient in matrix form:(Bm)(αv)(lv′) = Bαvm (2.15)(vm)(αv) =Vα,vm (2.16)(am)(lv) = avlm (2.17)Where in Bm, the row matrix labels are combination of frequency ans baseline and the column indexlabels are the combinations of multipole and frequency. we are writing the equation 2.14 in simple linearform:v= aB+n (2.18)This equation is essence of m-mode formalism. The reduction of measurement process to a simplelinear mapping in equation 2.20, which uses finite number of degree of freedom, allows us to apply signalprocessing to recover 21cm signal. For more details refer to [2]. the intensity mapping experiments havemainly three components: 21 cm signal which we are interested in, the foregrounds and instrumental noise.The covariance of the visibility is represented by:10C(αvm);(α ′v′m′) =<VmαvVm∗α ′v′ >= ΣBαvlm < a∗lmval′m′v′ > Bα ′v′∗l′m′ +< n∗αmvnα ′m′v′ > (2.19)this is the covariance between all the possible degree of freedom, baselines, frequency and m modesand hence this is a huge matrix to be solved. If we assume the sky signal to be isotrpoic, this matrix is dra-matically simplified. The autocorrelation of the spherical harmonics in 2.19 is uncorrelated in m index.Thismeans both the computation requirement and data storage requirement is reduced by great amount. In thematrix form, equation 2.19 can be written as:C = BCskyB∗+N (2.20)Where Csky signal can be split to 21cm signal and foreground signal Csky =C21 +C f . these foregroundsignals are roughly 105 times brighter than 21 cm signals. Removing these signals is the first data reduc-tion step. Mode Mixing complicates the foreground removal process. The CHIME beams are frequencydependent which means that the side lobes response vary from zero to maximum for the sources which areoff zenith. At zenith it is fine as it is maximum for all frequencies. Total signal is the sum of response ofthe main beam which is maximum for each frequency and frequency dependent structure due to side lobes.This frequency dependent structure imitates the 21 cm signal. The cartoon for mode mixing is shown inFigure 2.1.The Karhunen—Loe`ve Transform (KLT) is a technique used to extract weak signals from noise. Thistechnique works more accurately than fast Fourier transform when the signals buried under the noise arevery week. In our case, we want to extract the 21cm signals. It uses the signal and foreground covariancematrices and diagonalizes them simultaneously, generating the set of eigen-basis. Then we take the modeswhich satisfy the required signal to noise ratio. Figure 2.2 shows how we efficiently removed the foregroundsignal using KL transform without affecting 21cm signal largely. This plot is taken from [2].2.3 Calibration requirements for CHIMEThe primary beam response, LNA gain, cable delay and FLA gain will distort our measurements. In general,if we completely know our instrument, we just need to remove the bright foreground signals to see the BAOsignal. But these non-idealities results in distorting our signals. In this section we are going to reviewhow large these uncertainties can be before they matters. Our ability to separate signal and foregroundrequires a detailed knowledge of our instrument. [2] states that knowing the instrumental gain and beamresponse to 10−5 accuracy should be sufficient. Later it tests two particular forms of uncertainty to seeif the requirements are as stringent as 10−5. The gain fluctuations are modeled on feed by feed basis.These fluctuations are complex perturbations around gain of unity, with the variance σ2g . After applyingperturbations, feed input is represented asF ′i = (1+∆gi)Fi (2.21)where perturbation ∆g is complex Gaussian random variable with variance σ2g . The corrupt visibilities(vi j) are given by11Figure 2.1: Mode mixing due to frequency dependence of the beams. Side lobe response vary fromzero to maximum, depending on frequency. This frequency dependent structure in data imitates21cm signal. The blue line is the response of main beam at different frequencies, red line is theresponse of side lobes at different frequencies. Credits: Carolin Hofer.V ′i j = (1+∆gi)(1+∆∗g j)Vi j (2.22)Here the gains are fluctuating in time but are not fluctuating in frequency. We then generated the cor-rupted time-stream corresponding to the observations the true telescope would make. Figure.... shows thepower spectrum biases corresponding to each level of gain fluctuation with σg = 10%,1% and 0.1%. Wecan see that the bias becomes negligible for errors residuals of around 1%. Based on these simulations weneed to know the complex gain of each receiver to 0.3% within each minute in order not to bias the derivedpower spectrum by more than its statistical uncertainty.According to [2], we can recover unbiased power spectra when we can know the per-feed beam widthto 0.1%, and amplifier gains to be known to 0.3% within each minute. In the next section we will show thatthe 21 cm signal can be separated from the astrophysical foregrounds in a way which does not distort our12Figure 2.2: Process of the foreground removal from simulated sky. the top plot shows the simulatedsky maps for the 3 components: unpolarized foregrounds, polarized foregrounds and 21cm sig-nal. At the bottom, we show the sky maps for the 3 components mentioned above after KLfilter (foreground removal technique). Both the polarized and unpolarized signals are highlysuppressed and 21cm signal is largely unaffected. Plots from [2]13Figure 2.3: Biasing of power spectrum for complex gain perturbations with amplitude σg = 10%,1%and 0.1%. The foreground wedge is indicated by vertical dash lines, indicating that our methodcan clean foregrounds well below the foreground wedge. The bias is given as fraction of statisticalerror. regions shown in blue, where ratio is less than 1, indicates that the systematic errors aresub-dominant compared to statistical errors. Plots from [2]measurement of the underlying power spectrum.2.4 Thermal model for components in analog chainThe complex gain of each receiver depends on frequency and changes with the time. The gain variations aremainly due to the thermal changes in the environment. To account for the errors due to gain variations, weneed to determine the gain at real time and apply the gain corrections on the time scale faster than the rateat which sky signal changes.One of the technique to calibrate the instrument is to develop the thermal model for every componentin the analog chain. The complex gain (amplitude and phase) of every CHIME LNA are measured in thelaboratory at a single temperature. In addition, the gain of 100 LNAs were measured as a function oftemperature in a thermal chamber at UBC. These measurements were done by undergraduate students and14Figure 2.4: Power spectrum biasing for 10%, 1% and 0.1% shift from the fiducial E-plane width. Herewe can see that unknown fluctuations in the beam width of more than 0.1% give rise to significantpower spectrum biases. Plots from [2].grad student, Mateus Fandino. Figure 2.5 shows the mean amplitude and phase of LNA as a function oftemperature at 5 selected frequencies across the CHIME band. The amplitude varies between 0.003 and0.007 dB/K, with a slightly temperature-dependence to the slope, while the largest phase susceptibility isroughly 3.5e-3 radians/K at 800 MHz near room temperature.In addition, we measured the amplitude and phase of 100 FLA’s in the thermal chamber. Figure 2.6shows the mean amplitude and phase of FLAs as a function of temperature at 5 selected frequencies acrossthe CHIME band. The amplitude varies slightly more with temperature than do the LNAs, but the FLAsare located indoors so their overall temperature variation is less than the LNA’s. Like the LNA, their phaseis a non-linear function of temperature. We worked on a crude thermal model for the phase and amplitudeof the cables, shown in 2.7. Both the phase and amplitude of the cables are linear function of temperature.In the next step we initialize the thermal chain with the LNA, FLA and Cables all together. The phase andamplitude got the chain is shown in figure 2.8. It is clear from the figure that the phase and amplitude ofthe cables dominates over FLA and LNA. It is commonly known that the materials used to make a coaxialcable have positive thermal coefficient of expansion and that the electrical and physical length of the cableare directly related. It is obvious from this, that with the increase in temperature, the physics length and15Figure 2.5: The average complex gain, amplitude at left and phase at right, measured in 100 CHIMELNA’s for 5 frequencies across the CHIME band as indicated (in MHz). To meet our calibrationrequirements, we need to know the temperature of the LNA’s to within 0.85 Kelvin. Plots byMateus Fandino.Figure 2.6: The average complex gain, amplitude at left and phase at right, measured in 100 CHIMEFLA’s for 5 frequencies across the CHIME band as indicated (in MHz). The FLAs are in thermalcontrolled environment. To meet our calibration requirements, we need to know the temperatureof the FLA to 0.1K. Plots by Mateus Fandino.electrical length of the cable increases. The electrical length is also known as the phase length whichcorresponds to the number of wavelengths in cable at a given frequency. The main mechanism behind thechange of the phase length of these cables with temperature is due to change in the mechanical dimensionor the dielectric constant. The dimension of the conductor dimensions expand with increasing temperaturein a linear and predictable manner [8]. the central conductor expansion is the major reason for the affects inthe physical length of cables. The expansion of the outer conductor causes change in the mechanical forcewhich is acting on the dielectric material of the cables, which results in the change in density of dielectricmaterial. Hence the dielectric constant changes.16Figure 2.7: The average complex gain, amplitude at left and phase at right, measured for some CHIMEcables for 5 frequencies across the CHIME band as indicated (in MHz). Plots by Mateus Fandino.Figure 2.8: The amplitude (left) and phase (right) when we initialize the thermal chain with the LNA,FLA and Cables all together. The phase and Amplitude of the cables dominates. Plots by MateusFandino.17Chapter 3Complex Gain measurement of cables:Properties & Experimental setupIn this chapter we are going to discuss about the properties of 50m coaxial cables (LMR-400) used forCHIME. In section 3.2, we will be discussing some of the common round trip phase measurement schemesused to measure the phase of the signals . In section 3.3, we will discuss the experimental setup for the roundtrip phase measurement scheme that we used to measure the complex gain of the cables used in CHIME.3.1 Properties of LMR-400The 50m coaxial cables that carry RF signals from focal line to receiver hut in CHIME are LMR-400. Theseare low loss communication coaxial cables. There are total of 2048 - 50m coaxial cables used in CHIME.Some of the electrical and environmental specifications of these cables are described below:-1. The velocity of propagation of signals through these cables is 0.85c, where c is speed of light invacuum.2. The time delay of these cables is 3.92 nS/m.3. Operational range for these cables in -40/85 oC.4. These cables have impedance of 50 Ohms.These coaxial cables are passive and highly non-dispersive. This means that the time delay of cable isnot function of frequency. We measured the phase of the cables used in round loop test. The phase spectraof these 7 cables at an instant is shown in Figure 3.1. Each curve in the figure looks highly linear. Linearfit was performed on each curve in the Figure 3.1 and the linear fit is subtracted from the curve to get theresidue, shown in Figure 3.2. The slope of the linear fit is the delay of the cable at that instant, it is discussedin detail in Section 4. In general, the dispersion in the cables is due to the frequency dependence of skindepth. Theoretical calculations show, for a cable length of l at frequency f, the excess time delay due todispersion can be represented by:18Figure 3.1: Phase spectra at an instant for 7 round trip paths. The phase of the cables look highlylinear.τDispersive = 4.78(l100m)(10MHzf)ns (3.1)The total phase shift due to dispersion is represented theoretically by:φDispersive = 2pito( f +Lso2Lof 1/2) (3.2)where f represents frequency and t0 represents the signal propagation time from one end of the cableto the other. L0 is the ideal inductance mentioned above and Ls0 is a constant determined by the physicaldimensions of the inner and outer conductors as well as their conductivity. Based on equation 3.2, weperformed polynomial fit(adding f 1/2 term in addition to linear term) on each curve in the Figure 3.1 andthe fit is subtracted from the curve to get the residue, shown in Figure 3.3. At 600 MHz, the phase of cableA00 at an instant is 73.6085 rad, for linear and polynomial fit phase is 73.61007 rad and 73.6088 rad. Thedifference between raw phase and phase after polynomial fit @ 600 MHz is negligible. In [1], they gotthe value of coefficient for polynomial term ( f 1/2) in equation 3.2, 376.36to and we got 244.038to. Thedifference in values is due to the properties of the inner conductor, outer conductor and spacing materialused in cables for two cases.Attenuation of these cables is function of frequency, we will be discussing this in more detail in Section19Figure 3.2: Residual obtained by subtracting the Phase spectra at an instant for 7 round trip pathsshown in Figure 3.1 from their linear fit. The RMS residual is 0.025 ± 0.003. The fit coefficientsfor the linear fit are: (-24584±80)10−10 ( f − f¯ ) + (458665±2766)10−3, where f is frequency inHz and f¯ is 600x106 Hz. The horizontal line in the figure is at residual = 0.5 . In this thesis we will focus on the thermal modeling of complex gain of the cables as a function of timeand frequency.3.2 Delay measurement MethodsOne of the major challenge in radio interferometry is the calibration of the instrument. Path length variationsof the signals can be determined by monitoring the phase of the signals of known frequency that travels thepath. For this, it is necessary for the signal to travel in two directions, out from the signal generator (masteroscillator) and back, as the master oscillator provides the reference signal. Using the reference signal, we canmeasure the phase. This technique is referred to as the round trip phase measurement technique. Correctionof the measured phase changes can be applied using the hardware (using phase shifters) or in software byinserting the corrections in the data using correlator. Several round-trip schemes have been developed andsome of them are discussed below20Figure 3.3: Residual obtained by subtracting the Phase spectra at an instant for 7 round trip pathsshown in Figure 3.1 from their polynomial fit. The RMS residual is 0.023 ±0.003. The fitcoefficients for the quadratic fit are: (-24565±80)10−10 (z− z¯)2 + (-0.1204±0.0004)(z− z¯) +(458657±2764)10−3, where z is√ f . The horizontal line in the figure is at residual = 0.3.2.1 Swarup and Yang systemOn of the earliest round-trip phase measurement scheme was by Swarup and Yang. Figure 3.4 shows asystem based on the technique of Swarup and Yang. This is system for measuring the variations in the pathlength of the transmission line. The phase of the signal in the transmission line is monitored by measuringthe relative phase of the reflected component of the signal from a known reflection point at an antenna.The phase of the reflected component is compared with the reference signal. Since there might be manyreflections in the transmission line, to identify the desired component, modulated reflector is used.3.2.2 Frequency Offset round-trip schemeThe second scheme is frequency offset round-trip scheme. In this scheme, the circulators and directionalcouplers are used to separate the signals traveling in different directions and to suppress the signals from theunwanted directions by 20-30dB relative to the desired signals. The signals traveling in opposite directionsare at different frequencies so they can be separated from each other but theses two frequencies differ by verysmall amount. The oscillator at the antenna station with the frequency v2 is phase locked to the differencefrequency of signals at v1 and v1− v2. The signals with frequency v1 and v1− v2 travel from receiver station21Figure 3.4: System for measuring the variations in path length of the transmission line using Swarupand Yang’s round trip measurement scheme. Part of the signal is reflected from the known re-flection point at the antenna. The output of the synchronous detector is the sinusoidal functionof difference between the reflected component of signal and the reference signal. The position ofprobe for null (when signal phase is in quadrature) is the measurement of the phase of the signal.For the variation of ∆l in the length of transmission line, the probe position is shifted by 2∆l forthe null, where the phase of the signal is changed by 2pi∆lv1v . Figure from [9].to antenna station via transmission line. The frequency v2 is returned to the master oscillator location whichis at receiver station for phase comparison.3.3 Experimental setup for complex gain measurements.The round-trip phase measurement method is a technique to monitor the rf signal delay and attenuation inthe system. We used 14 Co-Axial cables and installed on CHIME by connecting them with adjacent cablesat the focal line, injecting a signal in the receiver room and using the stone correlator and ling laptop todetermine signal delay and attenuation.The setup for this experiment is shown in Figure 3.6. For the signals through the focal line there is one2m cable between splitter and outer bulkhead, two 50m cables to the focal line and back, there is one 2mcable from outer to inner bulkhead and one 2 m cable from inner bulkhead to correlator. For the referencechannel (shown in red) there is one 2m cable from splitter to inner bulkhead and one 2m cable from innerbulkhead to correlator. The two 50 m coaxial cables were connected using 1 ft cable at focal line. We usedFLA, LNA and terminator as noise source, shown in Figure 3.7. During this setup, cable trays were notcovered and the FLAs were not installed. For this experiment, we used stone correlator and data is collected22Figure 3.5: Phase locked scheme for the oscillator with frequency v2 at antenna station. Frequenciestransmitted from receiver station provides the phase reference to lock the oscillator at the antennaend. Here frequencies v1 and v1− v2 are transmitted from receiver station to antenna stationwhere they provide the phase reference to lock the oscillator. v1 and v2 are almost equal, theydiffer by small amount so that the signals belonging to these frequencies can be separated. Thesignal at frequency v2 is returned back to the central station for round-trip phase measurement.Figure from [9].using Ling laptop. The integration time(in FPGA) for this experiment is set to 10 minutes, i.e., 1 frame indata represents 10 minutes of data averaged.The stone correlator is 8-channel correlator based on Xilinx KC705 Eval board and McGill ADC board,shown in Figure 3.8 . This correlator is connected to a laptop, called ling, that runs CentOS (CommunityEnterprise Operating System) and is used for acquiring data. In Section 4 we will be discussing about thephase measurement and in Section 5 we will be discussing about the attenuation measurement of thesecoaxial cables.3.4 Geometrical layout of the cablesEach CHIME cylinder is 100m long, with 512 Co-Axial cables on each cylinder. The geometrical layout ofeach cable is different. The electrical length of these cables are temperature dependent. Location of the 7Co-Axial cables for this experiment were chosen in such a way that they cover wide range of run on focal23Figure 3.6: Round Trip Phase Measurement Scheme setup. There are total of 7 round trip cables. The50m cables that run to the amplifier on focal line, are disconnected and connected to the adjacent50m cable using 1 ft. cable. The plus sign in the figure represents the Dual-Polarized feeds onfocal line and red line is the reference cable, which is 2m SMA cable. The standard 2m cables areused in hut. There are bulkhead connectors at the inner and outer bulkheads at WRH. In general,the hut is cooled but during this experiment the cooling units were not in place.line and cable trays as shown in Figure 3.9. Cables from north and south end spend more time on focalline and less time on cable trays, whereas cables from middle of cylinder spend more time on cable traysand less on focal line. Portions of these cables are subjected to different temperature as the cable trays areunder the shade of cylinder whereas focal line is not. The main motivation behind this is to see if there issome grouping related to the properties of these cables. The cables in the south end are represented by thenumbering A00 and B00 for cylinder A and B respectively. The cables in north end are represented by A63and B63. The numbers mentioned in the figure represents the length of that cable on the focal line. Forexample, the cable from the focal line position A00, 10 m of that cable runs on focal line and rest 40m oncable tray.24Figure 3.7: Setup of the Noise source (FLA+LNA+Attenuator), 8 channel correlator and ling laptopfor round trip phase measurement scheme in West Receiver Hut.25Figure 3.8: Setup of 8 channel correlator (stone correlator) used in the experiment. The board in greenis Xilinx FPGA board and the board in red is McGill ADC board.Figure 3.9: Geometrical layout of the Co-Axial cables selected for this experiment. The cables in thesouth end are represented by the numbering A00 and B00 for cylinder A and B respectively.The cables in north end are represented by A63 and B63. These numbering are the focal lineposition of the cables, where the alphabet in the starting represents the cylinder followed by twodigit number representing the focal line position. The cable trays were not covered during thisexperiment, but they are covered now.26Chapter 4Delay Measurement & ModelingIn previous section we discussed about the round trip phase measurement scheme used to measure thecomplex gain of the coaxial cables used in CHIME. In this chapter we are going to discuss about the delaymeasurement of the coaxial cables. Our motivation for this analysis is to see if the air temperature is goodproxy for delay of these cables? Or if the delay of single cable represents the delay of other cables? Or thereis some grouping based on the location of cables on focal line?4.1 Delay MeasurementIt is commonly known that the materials used to make a coaxial cable have positive thermal coefficient ofexpansion and that the electrical and physical length of the cable are directly related. It is obvious from this,that with the increase in temperature, the physics length and electrical length of the cable increases.Here, delay (τ) is defined as the rate of change of the total phase shift with respect to angular frequency,τ =− dφdωthrough any medium, φ is total phase shift in radians, and ω is angular frequency in radians per unitsec, which is equal to 2pif, f is frequency in Hertz. Figure 4.1 represents the delay as function of time for 7channels. LMR-400 Co-Axial cables are used to connect the outer-bulkhead to feed. As per the data sheet,these cables have the time delay of 3.92 nS/m. The rf propagation delay varies with temperature. The ripplesin each curve are daily in length, these are probably related to the temperature. This variation amount toapproximately 1mm per oC, which is common mode, as the cables are closely matched in length. The cablesvary in physics length by as much as 8-10cm in 50m. Some of these variations are the variations with thetemperature which results in phase variation between channels. It can be seen that there is some constantdifference in the delay of each curve, that might be indicating that the cables are different in length.As discussed above, we are interested in the temperature dependent structure in the delay curve. Figure4.2 represents the scaled delay curve. For each curve in Figure 4.1, mean and variance is found and is usedto take out gain and offset for all days. Once scaled, curve looks very similar in shape, probably indicatingthat they might have similar temperature. To see how much scaled delay is varying with air temperature,27Figure 4.1: Measured delay as function of time for 6 cables used in this experiment. The ripplesin each curve are daily in length and are probably related to the temperature. There is gapbetween the delay of different cables, the gap seems to be constant in time. This gap/variationis due to the difference in the original length of different cables(this is not time/temperaturedependent, referred to as common mode structure) and temperature dependent change in lengthof cables(what we are interested in!). Here, A00 represents the position of cable at focal line, itcorresponds to south end of cylinder A and (1,4) represents the cross-correlation between channel1 and 4, where channel 4 is reference channel. Same legend is used for this whole analysis.scaled air temperature is plotted in the same figure. Air temperature data is obtained from DRAO.In figure 4.2, there is some structure in scaled delay curves which is missing in scaled temperature curve.To see this missing structure, the scaled delay curves are subtracted from scaled temperature curve, shownin Figure 4.3 . Clearly, scaled temperature curve is not the true representation of the delay curve.4.2 Delay Modeling & ResidualsOur motivation for this analysis is to see if we can define the delay of every cable by just measuring delayof one cable. If not, is there any grouping based on the geometrical layout of the cables? How much are wemissing from error budget in each case?The error budget is 0.3%, which corresponds to 0.6 picoseconds. We start by calculating change indelay (∆Di = Di− D¯i) for each cable and fitting it to air temperature. Figure 4.4 represents the residual28Figure 4.2: Scaled delay for 7 cables and scaled air temperature (bold green curve). For each curve inFigure 4.1, mean and variance is found and mean of each curve is subtracted and the variationsare scaled to a given unit, this is referred to as scaling. This scaling is done to take out gainand offset for all days. Once scaled, the curves look very similar and seem to follow the airtemperature very closely. The scaling units for temperature curve are: -0.2801 T + 1.2711, whereT is air temperature.(∆i = ∆Di−αiTi) when air temperature is taken as the proxy for the delay of cables. In this case, the errorbudget is missed by 58 picoseconds when there is sharp increase in temperature.In the next step, we want to see if there is any cable whom delay is representing delay of all the othercables? We tried to fit the ∆D of every cable to remaining 6 cables and picked one which is doing best basedon the residual (∆i = ∆Di−αi∆Dre f ). The best case is when the delay of cable A00 is taken as the reference.In this case, the error budget is missed by 18 picoseconds during daytime and 8 picoseconds during nighttime, shown in Figure 4.5. We tried to look for grouping based on which cylinder these cables are locatedat and position of these cable on focal line, but we didn’t find any obvious grouping.4.3 Non-obvious grouping for delayIn earlier section we didn’t find any obvious grouping based on the position of these cables. Based on Figure4.5 and after calculating correlation for delay, we found 2 non-obvious groups. One group is for cables withfocal line position: A63, A31,B00 and B63, second group is for cables with focal line position A00, A4829Figure 4.3: Residual plot obtained by subtracting scaled air temperature curve from the scaled delaycurve shown in figure 4.2. We can see that the deviations from the simple scaling of air tem-perature are very similar for all seven cable path. This is to see the missing structure in scaledtemperature curve. The horizontal line is at ∆i = 0.and B48. These grouping indicate that we did some mistake in mapping the focal line position to ADCinput. Figure 4.6 represents the residual plot for first group, where the change in delay (∆Di = Di− D¯i)of cable A63 is taken as reference and is fitted to the change in delay for cables with focal line position:A31, B00 and B63. For this group, the error budget is missed by 9 picoseconds and 1 picoseconds duringday and night time respectively. Figure 4.7 is the residual plot for second group, where the change in delay(∆Di= Di− D¯i) of cable A00 is taken as reference and is fitted to cables with focal line position: A48 andB48. For this group, the error budget is missed by 25 picoseconds and 1.5 picoseconds during day and nighttime respectively.30Figure 4.4: Residual (∆i) in pico-second, when air temperature is taken as reference. The horizontalline is at ∆i=0. Curves are following each other fairly well. It seems that cable temperatureis different from air temperature or there is some other ingredient other than air temperature todefine the cable temperature perfectly. The horizontal line is at ∆i=0.31Figure 4.5: Residual (∆i) in pico-second, when the delay of cable A00 is taken as reference. We triedto fit ∆D of every cable to remaining 6 cables and found the residual. Based on the residuals,cable A00 is doing best. The horizontal line is at ∆i=0.32Figure 4.6: Group 1: Residual in picoseconds, where the ∆D of cable with focal line number A63is taken as the reference. Linear fit for cables with focal line number A31, B00 and B63 isperformed and residual is found. The horizontal line is at ∆i=033Figure 4.7: Group 2: Residual in Pico-Seconds, where the ∆D of cable with focal line number A00 istaken as the reference. Linear fit for cables with focal line number A48 and B48 is performedand residual is found. The horizontal line is at ∆i=034Chapter 5Attenuation measurement & modelingIn previous chapter we discussed about the delay measurement and the residue for the cases when the air-temperature/delay of single cable was taken as the proxy for the delay of the other cables. Our next step isto see how to model the second part of the complex gain: Attenuation. In this chapter we will be discussingabout the attenuation in the cables and how the attenuation is function of temperature. We will start byseparating the linear component of the signal from the reflections. Followed by modeling the reflections andthe linear component of signal using air temperature. In the end we will discuss the residual, to see howgood did we model the attenuation of cables, knowing the allowed error budget is ±0.3%.5.1 Attenuation in CablesThe power loss in the cables is referred as attenuation. This power loss is in many ways: Dielectric loss,Resistive loss or due to different connectors in the signal path. Resistive loss in the coaxial cable arise duethe resistance of the conductors and results in heat being dissipated. Due to the different connections inthe signal path, there is the impedance miss match which leads to the reflections inside the cables. In otherwords, attenuation is the loss of transmission signal strength and is measured in decibels (dB)G j =Vi jViiHere, G j is the attenuation of the channel j, and is defined as the ratio of cross correlation of channel j withreference channel i and auto-correlation of reference channel. It is unitless quantity, and can be convertedto Decibels (dB) using 20log10(G j). Figure 5.2 represents the raw gain of 7 cables at 600MHz.The motivation of this analysis is to develop the thermal model for the attenuation of the cables. Ourfirst step is to separate the linear/direct component of the signal and the reflections of the signal. Figure5.2 shows the gain spectra (blue curve) for a single cable at an instant and a linear fit, representing linearcomponent of the signal in black curve.35Figure 5.1: Raw Gain in Decibels[dB] for 7 round trip paths at 600 MHz.There is constant differencebetween the gain of different cables, this might indicate that the cables are of slightly differentlength.Figure 5.2: Gain spectra of a single cable at an instance (blue curve) and a linear fit, representing linearcomponent of the signal in black line.36Figure 5.3: Standing wave spectra for 7 round trip paths at an instant. This figure is obtained bysubtracting the linear fit to the gain spectra from gain spectra for each cable and at an instant.The standing wave spectra is different for each channel. This means, we need to measure andmodel the reflection spectra of each cable separately. We will be using the gain amplitude inlinear scale from here.5.2 Reflections in cablesReflections occur where there is an impedance difference between systems. Some of the power is transmittedthrough the cable and some of the signal is reflected back through the cable. These reflections are due tothe connections between cables, there can be impedance mismatch which causes some of the transmittedsignal to be reflected. These reflections can be separated using a powerful mathematical tool, Fast FourierTransformation (FFT, delay to frequency spectra) and Inverse Fast Fourier Transformation (IFFT, frequencyto delay spectra). In this analysis, we are separating the direct transmitted signal component from thereflections in the cable. Later in the section we will see that these reflections are temperature dependent. Tomodel these reflections we need to separate them from direct component of signal.Linear fit is performed on raw gain spectra for each channel at every time frame and is subtracted fromthe raw gain to get the reflection spectra. Figure 5.3 represents the standing wave spectra of 7 channels forsingle time frame. We observe that the standing wave spectra of each round trip path is different. Hence, weneed to model the reflection part of attenuation for each cable separately. Figure 5.4 represents the reflectionspectra of a single channel and each curve is different time frame. Same is done for every channel and weobserve that the curves follows each other fairly well. This indicates, there is common mode structurepresent here, which is not what we are interested in. We are more interested in the temperature dependentstructure, which is not visible in this figure.To see the temperature dependent structure, the average of all curves in Figure 5.4 is subtracted fromevery curve. Same is done for other channels. In other words, we are removing the common mode structure37Figure 5.4: Standing wave spectra for single channel (channel 1) and each curve represent differenttime frame (temperature). The curves seem to have a similar shape, indicating common modestructure present. To see the temperature dependent structure in these curves, we need to subtractthe average of all the curves from every curve.Figure 5.5: The two curves represents the standing wave spectra of the coldest and the hottest time forchannel 1.38Figure 5.6: Spectra of temperature dependent structure in Figure 5.4 for a single channel. This figureis obtained by subtracting the average of all curves in Figure 5.4 from every curve in the Figure5.4. [Note: We are not using data from this plot to model attenuation]and keeping the temperature dependent structure. Figure 5.6 represents the spectra of these temperaturedependent structure for a single channel. There are total of 800 curves in this figure, each curve representdifferent time frame.Our next step is to see how we can model these curves using temperature. To see these structure asfunction of time, Figure 5.6 is sliced at every frequency, and is plotted in Figure 5.7. Here, each curverepresents different frequency and two thick black curves are scaled air temperature. The shape of each curvein the plot is following scaled air temperature curve but with some delay. As discussed in the introduction,2m cables are in vestibule and 50m cables are outside. So, there are two temperatures which are dominant,temperature of 2m cables and the temperature of 50m cables. The lag indicates that the temperature of twocables are different from each other.If we separate the reflections due to 50m cables and 2m cables, we can model them using the 50mcable temperature and vestibule temperature respectively. To separate these reflections, we used a powerfulmathematical tool, FFT (Fast Fourier Transformation). FFT is an algorithm that samples the signal overthe time and convert them to frequency components and vice versa (IFFT). We followed the followingprocedure:-1. The linear component is separated from raw gain spectra to get the reflection spectra. This is donefor each channel for every time frame. (Figure 11)2. Reflection spectra is fourier transformed to get the delay spectra and reflections due to 50m cable and2m cable are separated.3. Spectra for the both reflections is obtained(Figure 5.9 and Figure 5.10) and common mode is taken39Figure 5.7: Scaled deviation plot for all the reflections in the signal path for a single channel. Differentcurves represents different frequency in CHIME band. The bold black lines are scaled air tem-perature. The deviation curves seems to follow scaled air temperature curve fairly well but thereis some lag. The cables of different length(50m & 2m) are used at different location in everychannel. There is a possibility that these cables have different temperature. [Note: We are notusing data from this plot to model attenuation]out from each curve and is sliced at every frequency.Figure 5.9 and 5.10 represents spectra of attenuation due to reflections in short and long cables respec-tively. The 7 subplots in each figure represents 7 channels. There are 800 curves in each subplot representingdifferent time frames. All the curves in each subplot seem to follow each other. To see the temperature de-pendent structure, average of curves in Figure 5.9 is subtracted from every curve. Figure 5.12 representsthe scaled deviations for reflections due to long cables, these are the temperature dependent structure in theattenuation due to reflections in long cables. The bold black curve in plot represents scaled air tempera-ture. The shape of each curve in the plot seems to follow scaled air temperature curve very closely. Inearlier section we saw that the Delay curve for each cable is better representation of 50m cable temperature.Figure 5.11 represents the scaled deviations for reflections due to short cables. These short cables are 2mcables and are in vestibule. Clearly there is some lag between the scaled air temperature curve and reflectioncurves. This means that the air temperature is not true representation of the vestibule temperature. We needto monitor the vestibule temperature to get the model for attenuation due to reflections in short cables. Forthis analysis we took average of 10 curves in Figure 5.11 and used it as the proxy for vestibule temperatureto model these reflections.40Figure 5.8: Delay spectra of all the curves in Figure 5.4. The plot represents how we are separatingthe reflections.5.3 Thermal Model for AttenuationIn earlier section, first we separated the linear part (direct component) of the attenuation from the reflectionsand then we separated the reflections due to the short and long cables. In nutshell, we have the 3 componentsof attenuation to model:-1. linear component2. Reflections due to short cables3. Reflections due to long cablesThe linear component is temperature dependent, while the other two components are both frequencyand temperature dependent. Our next step is to model these three components of attenuation using airtemperature and proxy for vestibule temperature. For linear component(Glin = mf + c(t)), there are twopart to be modeled, slope(m) and Y-intercept(c). Figure 5.13 represents the slope of linear fit as function oftime for 7 channels. For the analysis, we took this to be a constant number for each channel. Figure 5.14represents the scaled y-intercept for 7 channels as the function of time. The curve in black is the scaled airtemperature and the other curves seems to follow scaled air temperature curve very closely. Similarly, thereflection terms are modeled using the air temperature and vestibule temperature for frequency. The firstthree panels in figure 5.15 represents the modeled and the raw gain for the three components discussed aboveand the bottom-right plot represents the difference in the thermal model and raw data for the 3 components41Figure 5.9: Spectra of attenuation due to reflections in short cables. The 7 subplots represents 7channels. There are 800 curves in each subplot, representing different time frame. All the curvesin each subplot seems to follow each other indicating common mode structure present.of gain mentioned above ,and the difference of raw gain and modeled gain, shown in blue.5.4 ResidualNext step is to see if we are doing under error budget. The error budget for attenuation is 0.3%. The modeledfractional change in gain (Gm(t)−Gm(t)Gm(t), Gm(t) is the modeled gain for single channel) is subtracted from rawfractional change in gain (Graw(t)−Graw(t)Graw(t), Graw(t) is the raw gain for single channel) and is plotted in Figure5.16. We are doing under the error budget during night time and we are off the limits when there is sharpchange in temperature.42Figure 5.10: Spectra of attenuation due to reflections in long cables. The 7 subplots represents 7channels. There are 800 curves in each subplot, representing different time frames. All thecurves in each subplot seems to follow each other indicating common mode structure present.43Figure 5.11: Scaled deviations for short cables for different frequencies and single channel. The curvein black and blue is the scaled air temperature. These short cables are in vestibule and it seemsthat there is lag between air temperature curve and other curves. This shows that air temperatureis not true representation of vestibule temperature/2m cable temperature and we need to monitorthe vestibule temperature to model these curves.Figure 5.12: Scaled deviations for attenuation due to reflections in long cables for single channel. Thecurve in black is the scaled air temperature and other curves are attenuation due to reflections inlong cables at different frequencies. All the curves seem to follow air temperature closely.44Figure 5.13: Slope(m) of linear fit (Glin = mf + c(t), f is frequency in MHz) as function of time for 7channels.Figure 5.14: Scaled y-intercept (c(t)-c(to) of linear fit as function of time for 7 channels. Curvein black represents the scaled air temperature, it seems that the scaled y-intercept follows airtemperature very closely and hence we used the air-temperature to model this component.45Figure 5.15: Thermal Modeling of each component of the attenuation for channel 1 at 600 MHz.(top-left) is the thermal model and the raw data of the linear part of attenuation, represented inblack and red color respectively. (top-right) is the thermal model and raw data for reflections inlong cables. (bottom-left) shows the thermal model and the raw data for the reflections in shortcables. (bottom-right) represents the difference in the thermal model and raw data for the 3components of gain mentioned above ,and the difference of raw gain and modeled gain, shownin blue.46Figure 5.16: Residual plot for attenuation at 600 MHz. The horizontal lines at 0.3% and -0.3% repre-sents the allowed error budget. We are doing under the error budget during night time.47Figure 5.17: Algorithm explaining how we are modeling the attenuation.48Chapter 6Future workIn this section we are going to discuss about the future work that can be done to calibrate for instrumentalerror by observing the point source in sky.As mentioned in section 2.3, we need thermal model for all the elements in analog chain. We alreadyhave a robust thermal model for LNA and FLA. In this thesis we discussed about an approach to develop thethermal model for cables. We used some cables already installed on CHIME to get the phase and amplitudeof those cables. We are doing under the error budget for attenuation during night time. The challenge is toseparate the reflections in cables in each channel to get a thermal model for attenuation. For phase, we stillhave to investigate more, as we didn’t find any obvious grouping between the phase of different cables. Thismight indicate that all the cables are working differently and we need to monitor the phase of each cableseparately. In section 4.3, we observed two non obvious groups. This might indicate that we did some errorin mapping the ADC input and focal line number.Our next step is to investigate the potential of a thermal model derived from sky data. The advantage ofthis approach is that individual thermal models can be created for each analog chain without the limitationof inter-component variability. Our first step is to extract the gains of each channel for a transit. These gainsare the combination of the gains of beams, amplifiers, cables and filters. By calculating the fractional changein gain for each channel, we can get rid of the beam gains. As the beams does not have time/temperaturevariability. The detailed approach to develop the thermal model is described below:1. Obtain the Visibilities for bright radio source(Cas-A/Cyg-A/Tau-A), preferably when the transit isduring night time. We made sure that the visibilities comprise of wide range of temperature.2. Solve for gains using Mateu’s gain solver algorithm for the well behaved channels in pathfinder dataset. If working on CHIME data, then use CHIMEcal data set to get the gains for each channel in CHIME,how to derive the gains, is described in the step below.PATHFINDER DATA: We derived gains from Cyg-A transit data and calculated the fractional changein gain. Amplitude fluctuations at four different frequencies for a particular channel as a function of outsidetemperature is shown in Figure 6.1. Circles denote measured values. Line denote the best-fit model fora linear regression of the amplitude fluctuation against temperature. Slope of line represents the thermal49coefficient at a particular frequency and a single channel. The thermal coefficient of -0.2 % / deg C isroughly what we expect for coaxial cables. Thermal coefficient of the amplitude fluctuations as a functionof frequency is shown in Figure 6.2. The dots in red represents the thermal coefficient when we used thewhole data set (55 days). We want to see if the ripples are constant with temperature/time, we divided thewhole dataset in two parts, each part with wide variety of temperature and did the analysis using these twodatasets and the thermal coefficients are represented in green and blue dots.CHIME: We used the ChimeCal dataset to get the gain for all channels. We calculate gain using firstand second eigenvalues and eigenvectors. One of the eigenvalue is significant for one polarization and othereigenvalue is significant for other polarization. Gain for feed i on CHIME is given by:Gi, f =√λ 0f |X0i, f |2+λ 1f |X1i, f |2Where, Gi, f is gain for feed i at frequency f and is given by square root of sum of product of firsteigenvalue(λ 0f ), square of first eigenvector(X0i, f ) and product of second eigenvalue(λ1f ) and square of secondeigenvector(X1i, f ).3. Next step is to construct the spatial temperature model for both LNA and FLA panel to estimate thetemperature for all the FLA and LNA.4. Subtracting the existing thermal models for FLA and LNA from the gains derived in step 2. Remain-ing gain is due to the cables and . Using the vestibule temperature and air temperature, develop the thermalmodel for the cables. This step requires more investigation.50Figure 6.1: Amplitude fluctuations at 633 MHz, 640 MHz, 644 MHz and 650 MHz (left to right, topto bottom) as a function of temperature for a single pathfinder channel. Circles denote measuredvalues. Lines denote the best-fit model for a linear regression of the amplitude fluctuation againsttemperature. The slope of the line is the the thermal coefficient at that particular frequency.51Figure 6.2: Thermal coefficient of the amplitude fluctuations as a function of frequency for channel53. The dots in red are when we did the above analysis using the whole data set (55 days).We want to see if the ripples are constant with temperature, so we divided the whole dataset intwo parts with wide variety of temperature and did the analysis using these two datasets and thethermal coefficients are represented in green and blue dots. The vertical red lines are at 30 MHzseparation. 30 MHz ripples are characteristic of the beams and we are not seeing 30MHz ripplehere, hence we don’t have beam effects here.[This is from pathfinder dataset]52Figure 6.3: First and second Eigenvalues for 2 frequencies. This is CHIME Cyg-A transit data.53Figure 6.4: Thermal coefficient of the amplitude fluctuations as a function of frequency for channel10. The vertical red lines are at 30 MHz separation. 30 MHz ripples are characteristic of thebeams. We can improve the signal to noise ratio by performing Gaussian fit to the eigenvalues.In Figure 6.3, we can see that each curve has some noise, doing the Gaussian fit on each curve canincrease the signal to noise ratio in this plot.[This is from CHIME dataset, Cyg-A night transit]54Bibliography[1] Dispersion in Coaxial Cables, Steve Ellingson Long Wavelength Array Technical.Report 136 (2008)[2] All-Sky Interferometry with Spherical Harmonic Transit Telescopes, Shaw et al. 2013, AstrophysicalJournal, 781, 57[3] Adam G. Riess, Alexei V. Filippenko, Peter Challis, Alejandro Clocchiatti, Alan Diercks, Peter M. Gar-navich, Ron L. Gilliland, Craig J.Hogan, Saurabh Jha, Robert P. Kirshner, B. Leibundgut, M. M.Phillips,David Reiss, Brian P. Schmidt, Robert A. Schommer, R. ChrisSmith, J. Spyromilio, Christopher Stubbs,Nicholas B. Suntzeff, andJohn Tonry. Observational evidence from supernovae for an accelerat-ing uni-verse and a cosmological constant.The Astronomical Journal,116(3):1009, September 1998.[4] S. Perlmutter, G. Aldering, G. Goldhaber, R. A. Knop, P. Nugent, P. G.Castro, S. Deustua, S. Fabbro,A. Goobar, D. E. Groom, I. M. Hook,A. G. Kim, M. Y. Kim, J. C. Lee, N. J. Nunes, R. Pain, C.R. Pennypacker, R. Quimby, C. Lidman, R. S. Ellis, M. Irwin, R. G. McMahon,P. Ruiz-Lapuente, N.Walton, B. Schaefer, B. J. Boyle, A. V. Filippenko, T. Matheson, A. S. Fruchter, N. Panagia, H. J. M.Newberg, W. J. Couch, and The Supernova Cosmology Project. Measurements of Ω and Λ from 42high-redshift supernovae.The Astrophysical Journal, 517(2):565, June 1999.[5] N. Jarosik, C.L. Bennett, J. Dunkley, B. Gold, M.R. Greason,M. Halpern, R.S. Hill, G. Hinshaw,A. Kogut, E. Komatsu, D. Lar-son, M. Limon, S.S. Meyer, M.R. Nolta, N. Odegard, L. Page,K.M.Smith, D.N. Spergel, G.S. Tucker, J.L. Weiland, E. Wollack,and E.L.Wright. Seven-year Wilkin-son Microwave Anisotropy Probe (WMAP)Observations: Sky Maps, Systematic Errors, and Basic Re-sults.TheAstrophysical Journal Supplement Series, 192:14, 2011.[6] S. Dodelson.Modern Cosmology. Academic Press, 2003.[7] E.J. Eisenstein. Dark energy and cosmic sound.New Astronomy Re-views, 2005.[8] K.Czuba, D.Sikora Temperature Stability of Coaxial Cables, Physical Aspects of Microwave and RadarApplications, Vol.119(2011)[9] Thompson A.R., Moran J.M., Swenson G.W. (2017) System Design. In: Interferometry and Synthesisin Radio Astronomy. Astronomy and Astrophysics Library. Springer, Cham55[10] D. J. Eisenstein, H.-j. Seo, and M. J. White, “On the Robustness of the Acoustic Scale in the Low-Redshift Clustering of Matter,”.56
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Complex gain modeling of CHIME’s coaxial cables Guliani, Sidhant 2018
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Title | Complex gain modeling of CHIME’s coaxial cables |
Creator |
Guliani, Sidhant |
Publisher | University of British Columbia |
Date Issued | 2018 |
Description | CHIME is a new radio interferometer located at the Dominion Radio Astrophysical Observatory (DRAO) in Penticton, BC. The primary goal of CHIME is to constrain the dark energy equation of state by measuring the expansion history of the Universe using the Baryon Acoustic Oscillation (BAO) scale as a standard ruler. CHIME consists of 4 cylindrical reflectors, each populated with 256 dual-polarization antennas along its focal-line. Prior to digitization, each signal chain consists of a low noise amplifier, 50m of coaxial cable, and a filter amplifier. In order to obtain accurate interferometric imaging, we need to determine the relative complex gain (amplitude and phase vs. frequency) of each analog chain to 0.3%. The complex gain of each receiver depends primarily on temperature. This thesis discusses efforts to construct a thermal model of the CHIME’s coaxial cables that will allow us to meet our calibration requirements. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2018-08-31 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0371866 |
URI | http://hdl.handle.net/2429/67041 |
Degree |
Master of Science - MSc |
Program |
Physics |
Affiliation |
Science, Faculty of Physics and Astronomy, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2018-11 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
AggregatedSourceRepository | DSpace |
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