Calibrating CHIME, A New Radio Interferometer to Probe Dark Energy
Laura B. Newburgh, Graeme E. Addison, Mandana Amiri, Kevin Bandura, J. Richard Bond, Liam Connor, Jean-François Cliche, Greg Davis, Meiling Deng, Nolan Denman, Matt Dobbs, Mateus Fandino, Heather Fong, Kenneth Gibbs, Adam Gilbert, Elizabeth Griffin, Mark Halpern, David Hanna, Adam D. Hincks, Gary Hinshaw, Carolin Höfer, Peter Klages, Tom Landecker, Kiyoshi Masui, Juan Mena Parra, Ue-Li Pen, Jeff Peterson, Andre Recnik, J. Richard Shaw, Kris Sigurdson, Michael Sitwell, Graeme Smecher, Rick Smegal, Keith Vanderlinde, Don Wiebe
CCalibrating CHIME, A New Radio Interferometerto Probe Dark Energy
Laura B. Newburgh a , Graeme E. Addison b , Mandana Amiri b , Kevin Bandura c ,J. Richard Bond de , Liam Connor df , Jean-Fran¸cois Cliche c , Greg Davis b ,Meiling Deng b , Nolan Denman f , Matt Dobbs c , Mateus Fandino b , HeatherFong b , Kenneth Gibbs b , Adam Gilbert c , Elizabeth Griffin b , Mark Halpern b ,David Hanna c , Adam D. Hincks b , Gary Hinshaw b , Carolin H¨ofer b , PeterKlages fg , Tom Landecker h , Kiyoshi Masui be , Juan Mena Parra c , Ue-Li Pen d ,Jeff Peterson i , Andre Recnik f , J. Richard Shaw d , Kris Sigurdson b , MichaelSitwell b , Graeme Smecher c , Rick Smegal b , Keith Vanderlinde fa , and DonWiebe ba Dunlap Institute for Astronomy & Astrophysics, University of Toronto, 50 St George St, Toronto,ON, M5S 3H4, Canada b Department of Physics and Astronomy, University of British Columbia, 6224 Agricultural Rd.Vancouver, V6T 1Z1, Canada c McGill University, 3600 University St, Montreal, Canada d CITA, 60 St George St, Toronto, ON, M5S 3H8, Canada e Canadian Institute for Advanced Research, CIFAR Program in Cosmology and Gravity, Toronto,ON M5G 1Z8 f Department of Astronomy & Astrophysics, University of Toronto, 50 St George St, Toronto, ON,M5S 3H4, Canada g IBM Canada h National Research Council Canada, Dominion Radio Astrophysical Observatory, Box 248, PentictonBC V2A 6J9 Canada i McWilliams Center for Cosmology, Carnegie Mellon University, Department of Physics, 5000 ForbesAve, Pittsburgh PA 15213, USA
ABSTRACT
The Canadian Hydrogen Intensity Mapping Experiment (CHIME) is a transit interferometercurrently being built at the Dominion Radio Astrophysical Observatory (DRAO) in Penticton,BC, Canada. We will use CHIME to map neutral hydrogen in the frequency range 400 – 800 MHzover half of the sky, producing a measurement of baryon acoustic oscillations (BAO) at redshiftsbetween 0.8 – 2.5 to probe dark energy. We have deployed a pathfinder version of CHIME that
Send correspondence to [email protected] a r X i v : . [ a s t r o - ph . I M ] J un ill yield constraints on the BAO power spectrum and provide a test-bed for our calibrationscheme. I will discuss the CHIME calibration requirements and describe instrumentation we aredeveloping to meet these requirements. Keywords:
1. INTRODUCTION
CHIME is a new radio interferometer currently being built at the Dominion Radio AstrophysicalObservatory (DRAO) that will operate between 400 – 800 MHz. The primary scientific goal ofCHIME is to constrain the dark energy equation of state by measuring the expansion history ofthe Universe through its impact on large-scale structure. We will measure the BAO comoving ∼
150 Mpc scale in a redshift range 0 . < z < .
5, which encompasses the epoch when theΛCDM model predicts dark energy began to dominate the energy density of the Universe. Toobtain sufficient survey volume for this measurement, we must map large-scale structure overhalf of the sky. To achieve this goal, we will use the neutral hydrogen 21 cm emission to tracethe large-scale matter distribution over our required redshift range across the entire skyaccessible from the DRAO.The CHIME instrument consists of five adjacent f / .
25 cylindrical dishes, each 20 m wideby 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 towest. The frequency-dependent resolution of the instrument is fixed by its longest baseline to be0.26 ◦ – 0.52 ◦ . This resolution is optimized for measurements of BAO features in the hydrogenmaps within our redshift range.The sky signal is focused by the cylindrical dishes onto dual-polarization feeds fixed to thefocal line of each cylinder. The resulting signals are amplified by ambient-temperature first-stage low-noise amplifiers (LNAs) attached directly to the output of the feeds. The signalis then carried down either coaxial cable or a radio frequency-over-fiber system to a secondset of amplifiers and a 400 – 800 MHz bandpass filter. As shown in Figure 1, the signals arethen digitized, channelized into 1024 frequency bands, and correlated. The feeds are spaced ∼
30 cm along the focal line, in a highly redundant pattern. The salient features of CHIME aresummarized in Table 1.As described in a companion paper in these proceedings, we have built and are commis-sioning the CHIME Pathfinder consisting of two adjacent cylindrical dishes, each 20 m wide by37 m long. The Pathfinder serves as a test-bed for developing instrumentation and as a platformfor testing calibration and analysis techniques. It will have sufficient sensitivity to measure BAOat low redshift.Due to the bright foreground signals present in the radio sky, accurate calibration techniquesare critical if 21 cm intensity mapping is to succeed. In these proceedings we describe thecalibration requirements for CHIME and our plans to meet them. We illustrate these planswith early Pathfinder commissioning data and with end-to-end simulation studies.igure 1: Schematic of the five cylindrical dishes and data processing which comprise the CHIMEinterferometer. There are 256 dual-polarization feeds on each cylinder whose signals are chan-nelized into ∼ ◦ (cid:48) N, 119 ◦ (cid:48) W)Number of inputs 2560Frequency range 400 – 800 MHzFrequency resolution 0.39 MHzWavelength range 75 – 37 cmRedshift range z = 2.5 – 0.8Epoch 11 – 8 GyrE-W FOV 2.5 ◦ – 1.3 ◦ N-S FOV ∼ ◦ about zenithAngular resolution 0.52 ◦ – 0.26 ◦ Spatial resolution 10 – 50 MpcTable 1: The salient features of the CHIME instrument.
2. CALIBRATION OVERVIEW
CHIME is a spatial interferometer: the data we record are correlations between signals fromthe feeds in the instrument. In this section, we present an overview of the quantities we mustcalibrate for CHIME, as well as the requirements on the precision of the calibration.
The main task of CHIME calibration is to convert measured voltages from the correlator intosky maps with true sky values. We can write the frequency ( ν ) dependent response of feed i ton incident electric field ε a ( ˆn , ν ) as F i ( φ, ν ) = (cid:90) d n g ai ( ν ) A ai ( ˆn ; φ, ν ) ε a ( ˆn , ν ) e πi ˆn · u i ( φ,ν ) , (1)where g ai ( ν ) is the frequency-dependent complex receiver gain and A ai ( ˆn ; φ, ν ) is the antennapattern of feed i and polarization state a , and we implicitly sum over the two polarization states.The antenna pattern includes telescope pointing; because CHIME is a transit interferometer thisis which is just the Earth’s rotation angle φ and can simply calculated from sidereal time, and Iwill drop the explicit dependence. The measured voltage is referred to as a visibility and is givenby the time-averaged correlation between the response of two feeds i and j ( i = j is referred toas an auto-correlation, and i (cid:54) = j is a cross-correlation) and includes instrument noise n ij : V meas ij = (cid:104) F i F ∗ j (cid:105) = (cid:90) B ij (cid:104) ε ( ˆ n , ν ) · ε ( ˆ n (cid:48) , ν ) (cid:105) d ˆn + n ij (2)This can be decomposed into maps of the Stokes parameters T ( ˆn ), Q ( ˆn ), U ( ˆn ), V ( ˆn ), as V meas ij = (cid:82) d ˆn (cid:104) B Tij ( ˆn , ν ) T ( ˆn , ν ) + B Qij ( ˆn , ν ) Q ( ˆn , ν ) + B Uij ( ˆn , ν ) U ( ˆn , ν ) + B Vij ( ˆn , ν ) V ( ˆn , ν ) (cid:105) + n T,Q,U,Vij (3)where the beam transfer functions, B T,Q,U,Vij encode all of baseline, polarization, and antennapattern information about the instrument. For more details on the formalism, see Shaw et al. The BAO signal is a fluctuation in the temperature map only, but we must calibrate the fullpolarization dependence in the beam transfer function to prevent polarized foreground signalfrom leaking into the BAO temperature signal of interest to CHIME. Thus, to convert measuredvoltages into maps, we must: • Determine the antenna pattern of each feed and polarization, A ai ( ˆn ; ν ), at each frequency. • Characterize the complex gain of each receiver as a function of time and frequency, g i ( t, ν ). • Account for cross-talk in the instrument over a range of lag times; schematically ˜ F i ( t ) = (cid:80) j α ij F j ( t − τ ij ). • Characterize the instrument noise, n ij ( t ) in Kelvin. CHIME must produce sensitive, foreground-cleaned maps of the sky on co-moving scales betweenthe instrument resolution and a few times the BAO scale of 148 Mpc. To measure BAO fromthe data power spectrum, we require that the systematic errors from propagating foreground fil-tering, calibration, and other effects not dominate the statistical error. This requirement drivesthe CHIME calibration accuracy requirements, but is made particularly difficult to achieve bythe presence of extremely bright astrophysical foreground signals, notably synchrotron emissionfrom our own and external galaxies. As measured at 408 MHz by Haslam et al., foregroundrightness can be as high as ∼
700 K near the Galactic center with typical mean signals of 10 –20 K at high galactic latitudes. Synchrotron emission typically falls with increasing frequencyas ν − . , yielding high-latitude brightnesses of ∼ ∼ to measure the 21 cm fluctuations accurately. Fore-ground filtering is made tractable by the spectral smoothness of the synchrotron emission, whicharises from cosmic-ray electrons spiraling in Galactic magnetic fields.Setting the calibration requirements for CHIME is a complex task that requires end-to-end simulations of the CHIME experiment, from simulating visibilities with realistic inputs tofiltering and analyzing the data to probe the effects of various instrument properties on thefidelity of the simulated data streams. We are undertaking a comprehensive simulation programto aid all aspects of the experiment, from hardware design choices to the setting of calibrationrequirements. The requirements we summarize below highlight the most critical aspects ofCHIME’s calibration, informed largely by our simulation program. Further details may befound in Shaw et al.
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Beam response - The requirements for the calibration precision of the frequency- andpolarization-dependent antenna pattern are driven by two effects that introduce foregroundspectral structure: (1) frequency dependence in the antenna beam converts angular structure inthe foreground signal into spectral structure in any given visibility, and (2) polarized foregroundsignals undergo Faraday rotation in the Earth’s ionosphere and the interstellar medium whichintroduces spectral structure in our visibilities. Since the magnitude of these effects dependson aspects of the foreground signal that are poorly known, we make pessimistic assumptionswhen setting these requirements. We have an ongoing simulation program to study the effectsin detail and we note a key result here. In an end-to-end simulation where we perturb thefull-width-half-maximum (FWHM) of each feed beam by a fixed amount that varies randomlyfrom feed to feed, we find that each beam FWHM must be known to 0.1% in order not to biasthe derived power spectrum by more that its statistical uncertainty. We continue to study theeffects of other beam pattern errors in a similar manner. We discuss our plans for meeting theserequirements in § Complex gain - Establishing relative complex gain (amplitude and phase) as a function oftime is required to achieve relative brightness accuracy and to reliably combine multiple visibil-ities. We have propagated random errors in the gain through the analysis pipeline (includingforeground filtering and power spectrum estimation), and conclude that random variations mustbe less than 1% on 60 s time scales. This level limits bias in our power spectrum estimation to ∼
10% of the statistical uncertainty. This applies independently to the real and imaginary partsof the gain, and is hence a de facto requirement on gain phase. We discuss our plans for meetingthis requirement in §
4. We will not include calibration to obtain the absolute sky brightnessfor CHIME because it is not critical to achieve our science goals. This is because the we willmeasure fluctuations in brightness generated by BAO, and so only differences in power betweenspatially separated regions of the sky are relevant for BAO power spectrum constraints.
Cross-talk - Coupling between channels in the detection path (cross-talk) can occur betweencables, between feeds on the cylinder, and within the digitizer and correlator boards. Cross-talkrom cables between the focal line and the correlator have values of better than −
120 dB (for thelow-loss shielded coaxial cables on the focal line) and better than −
70 dB (for additional cablesin the electronics building). Laboratory measurements of cross-talk within the digitizer boardsproduce estimates of ∼ −
50 dB. We are working to understand and constrain the cross-talkbetween feeds on the cylinder using a portable vector network analyzer (VNA) to inject a signalinto one feed and measure the response of neighboring feeds.Cross-talk in our cross-correlation measurements adds both noise and signal with coefficientswhich are typically stable in time. The noise comes from cross-talk between the cross-correlationbaselines and the auto-correlations, and appears in the cross-correlation time streams as a DCoffset. Cross-talk between different cross-correlation products will leak signal from one baselineinto the other. This will not generically be an additive constant offset, but will appear as signalat different phases, corresponding to non-main-beam pointing on the sky. These coefficients willhave to be measured, then estimated and extracted from the data during analysis of the fulldata set.
Instrument Passband - The process of foreground filtering requires knowledge of oureffective frequency bandpass to a part in 10 . This requirement mixes knowledge of our gainand of our antenna response as a function of frequency. It will be met by demanding that thebrightest foreground regions in our maps simultaneously have a smooth frequency spectrumto a part in 10 . This must include the two previously mentioned effects (mode mixing andFaraday rotation) which introduce foreground spectral structure that must be accounted forin the filtering process. The degree to which beam effects complicate this approach is beingactively studied at this time, so it is difficult to specify a bandpass requirement independent ofa beam knowledge requirement.We will describe the ongoing efforts to map the CHIME beams and understand the com-plex gain of CHIME throughout the rest of these proceedings. In § §
3. CHIME BEAMS
The beam from an individual feed element on the CHIME cylinder will extend from the northernhorizon to the southern horizon, with an east-west width determined by the cylinder diameter(1.1 ◦ – 2.2 ◦ from 800 – 400 MHz). We expect the polarization response to vary through the beamshape. Because a single recorded voltage contains the integrated signal from the large CHIMEbeam, it is particularly challenging to precisely calibrate the CHIME polarized beam shape. Wecan use transits of bright sources for basic characterization, and are developing a method ofpulsar holography to address these challenges and achieve the required 0.1% precision.igure 2: Shown is the fitted FWHM of the CHIME beam from a Cas A transit as a function offrequency in part of the CHIME instrument bandwidth. Data for the E-W (‘EP’) oriented beamand N-S (‘SP’) is in black. A 30 MHz ripple is seen in the data at a frequency consistent witha standing wave between the cylinder and the feed ground plane. Simulations of the CHIMEoptics with the ground planes were performed, and the resulting FWHM behavior matches theshape of the modulation (red and blue curves). We use daily transits of bright sources (CasA, CygA, TauA) through the CHIME beam to assessmain-beam size and pointing as a preliminary step towards fully characterizing the instrumentbeam. We fit a Gaussian profile to the transit data to compute its FWHM and the peakresponse location in right ascension at the declination of the source. The fitted FWHM valuesfrom a single auto-correlation channel for all frequencies from a CasA transit are shown inFigure 2. There is a ripple at a frequency of ∼
30 MHz, corresponding to the characteristicfrequency of a standing wave with wavelength 10 m, approximately twice the distance betweenthe focal line and the cylinder. These features are repeatable, suggesting that this is due to atrue standing wave: some of the signal is reflecting off the ground planes of the feeds, bouncingoff of the cylinder and back up to the feeds, and being detected again as signal. We modeledthis system with the reflector modeling software GRASP ∗ and recovered a spectrum with thesame peak locations seen in the data. These fits are included with the FWHM data in Figure 2.The declination-dependent location of the maximal response for different sources yields theorientation of each cylinder from the source transit data. Using source transits at differentdeclinations, the reconstructed pointing is consistent with the 2 ◦ offset between grid north and ∗ elestial north at the DRAO. This 2 ◦ offset is a relic of the surveyor’s usage of grid north whensighting the cylinder orientations and will not impact the pathfinder science result. Transit data from bright, constant sources will not provide polarized beam response measure-ments at the required precision because it is difficult to disentangle the signal from other astro-physical sources, particularly those located in the far sidelobes. Pulsars provide a sky subtractedsignal upon differencing ‘pulsar on’ and ‘pulsar off’ times. We will simultaneously measure pul-sars with the stationary CHIME instrument and the tracking 26 m telescope at the DRAOfacility to precisely map the polarized response of the CHIME beam. We attach a CHIME feedand amplifiers near the focus of the equatorially mounted 26 m diameter dish. It has a focalratio of 0.3, and we expect noise figures around 100 K. Using a CHIME feed on a telescopewith slightly larger focal ratio may lead to greater spillover and an increased thermal noisefloor, which may be mitigated with a small cone around the feed. 100 K noise temperatures areassumed for all calculations in this section.The signal from the 26 m telescope will be fed directly to the CHIME correlator to becorrelated with all of the CHIME channels. We refer to this measurement as ‘pulsar holography’.The technique of radio holography is a well-established method of mapping out a telescope’s far-field antenna pattern. In this simplest case, radio holography requires two telescopes: a referenceantenna that tracks a calibrator source (in this case the 26 m telescope) and another whose beamis to be calibrated (in this case CHIME). Pulsar holography facilitates accurate beam mappingbecause we only measure correlated signals between the two telescopes: the pulsar and anybackground sky common to CHIME and the 26 m telescope’s beam. We difference ‘pulsar on’times and ‘pulsar off’ times to remove the non-pulsed sky signal, leaving a clean measurementof the pulsar alone on fast time scales commensurate with the pulse period (of order 100 ms)because neither the sky nor RFI varies on such short timescales.Since CHIME is a transit interferometer that sees each visible source once per day, in practicewe will track pulsars in the range of ± ◦ of the meridian with the 26 m as they drift throughCHIME’s beams. This will generate a cross-correlation time-stream between the 26m and eachof the CHIME feeds, as well as a 26 m auto-correlation time stream. Together, they allow us tosolve for the CHIME beams using the formalism we present below.Mapping the CHIME beams is done by solving for the Jones matrix for each dual-polarizationCHIME feed, at every frequency, for as many points on the sky as possible given the spatialdistribution of pulsars and the signal-to-noise of the instrument. A single transit of the brightestpulsar should allow us to learn about our primary beam to better than 1% at that declination.In order to get a global beam fit we will need to find an optimal observing strategy in which weobserve each pulsar for multiple transits. In addition, the transit of a single pulsar yields onlyone slice of the CHIME beam at the declination of the pulsar. To fill out the rest of the north-south beam, we must repeat this measurement for pulsars at a range of declinations. Pulsars arehighly linearly polarized ( ∼ − e s ) to the measuredelectric field ( e m ): e m = Je s (4)The components of the electric field vectors are described by the two polarization states, inour case linear polarization (e X and e Y ), and J is a 2 × V ij = (cid:104) e m i e m † j (cid:105) = J i (cid:104) e s e s † (cid:105) J † j (5)Noting that (cid:104) e s e s † (cid:105) is just the true sky visibility (‘coherency matrix’: P ( t )), we can write theholographic response of the CHIME ×
26m cross-correlation as V hol ( t ) = J i (ˆ n ( t )) P ( t ) J † (ˆ n ( t )) + N hol ( t ) (6)where J is the known, stable Jones matrix of the 26 m telescope, N hol ( t ) is the measurementnoise covariance matrix, and J i (ˆ n ) is the Jones matrix of a CHIME antenna i as a function ofposition on the sky ˆ n . The latter is the quantity we want to solve for using the cross-correlationmeasurement between the 26 m and CHIME. Assuming a known J , we can use the measuredautocorrelation from the 26 m: V ( t ) = J P ( t ) J † + N ( t ) (7)to rearrange and find an estimate for the pulsar’s polarization matrix ˆ P ( t ) = J − V ( t ) J †− which we can insert into Equation 6 to obtain,ˆ J i (ˆ n ( t )) = V hol ( t ) J †− (cid:104) ( J − V ( t ) J †− (cid:105) ) − (8)= V hol ( t ) V ( t ) − J (9)Because the noise covariance term has been neglected, this estimator is suboptimal. Althoughit is beyond the scope of this paper, one can show the bias introduced by the non-zero mean of N ( t ) can be estimated.Using Equation 6 we simulate a holographic visibility time stream and reconstruct a CHIMEbeam given a set of pulsar observations. This simulation was performed assuming a beam takenfrom Shaw et al. and modeling the pulse-to-pulse flux variation as a log-normal distribution togenerate the 26 m auto-correlation timestream and the cross-correlation. We use these simulatedigure 3: Left:
Pulsar tracks through a simulated CHIME beam for 10 of the brightest pulsarsfrom the ATNF pulsar catalog.
Right:
The extrapolated CHIME beam from pulsar measure-ments, assuming a smooth and symmetric profile. A prior for the true profile will be informedfrom optics simulations. An example of an interference pattern for a source (CasA) between the26 m and CHIME is shown in Figure4.time streams to find the Jones matrix via Equation 9. We used the brightest sources fromthe ATNF pulsar catalogue which gave us a set of tracks through our beam at differentdeclinations. The tracks and resulting beam (assuming it is both smooth and symmetric) isshown in Figure 3. Work is currently underway to fit these tracks with a set of basis functionsto parametrize the full beam shape. The choice of basis functions and the fit will require bothpulsar measurements and some guidance from optical modeling.The above calculations assume a known Jones matrix for the 26 m telescope. The 26 mtelescope is equatorially mounted such that the telescope rotates with parallactic angle as ittracks the source in the sky. Equatorial tracking makes calibrating the polarized response ofthe 26 m difficult. This is because the least error-prone polarization calibration is derived fromtracking polarized sources as they rotate with parallactic angle, and fitting for instrumentalpolarization as the term that is constant with parallactic angle rotation. To properly calibratethe 26 m, we will require a large telescope which is alt-azimuth mounted. We plan to usethe 46 m dish at the Algonquin Radio Observatory (ARO) to perform a very long-baselineinterferometric (VLBI) measurement. We will first calibrate the alt-az 46 m telescope with astable, bright pulsar. We then perform the VLBI between the 26 m DRAO telescope and the46 m ARO telescope. The solution for obtaining the 26 m calibration from the ARO telescopeigure 4: Interference fringes between the tracking 26 m telescope at DRAO and CHIME asCasA transits through the CHIME beam, at ν =438 MHz.is identical to the expressions above for the 26 m × CHIME beam calibration. There are a fewadditional complications since we cannot directly cross-correlate the two VLBI telescopes, andwe must include the effect of the atmosphere, particularly Faraday rotation as the source photonspropagate through the Earth’s ionosphere.We began pulsar calibration measurements between the 26 m and CHIME in May 2014.Analysis of the signal-to-noise from the measurements and initial beam measurements frompulsars is underway, but we began by tracking CasA with the 26 m as it transited throughthe CHIME beam. The first fringes from this transit are shown in Figure 4 for frequency ν = 438 MHz.
4. GAIN DETERMINATION
The complex gain of each receiver depends on frequency and changes with time. Gain variationsare typically attributed to 1 /f noise in the amplifiers and to thermal changes in the environment.To mitigate systematic errors, we must determine and apply gain corrections on time scales fasterthan the rate at which the sky signal changes. In this section we describe various techniques weplan to use for gain calibration on CHIME, including laboratory characterization, broadbandnoise injection, switched sky sources (pulsars), and redundant baseline modelling. We have measured LNA gain of several CHIME amplifiers in a temperature-controlled chamberto find their thermal susceptibility over the range −
20 C to +50 C. The results for several ofthese amplifiers indicate that the typical amplifier gain varies by about − Using an injected source to measure relative receiver gains through time is a well-establishedpractice in radio astronomy. We have implemented a Broadband Injection Signal (BIS) forgain calibration on the pathfinder across the full CHIME bandwidth. A description of theBIS apparatus, the analysis technique, and preliminary results from tests on the pathfinder arepresented in this section.
A diagram of the setup for injecting the broadband calibration signal is shown in Fig. 5. Insidea screened room a switched noise source is installed. The output of this source is filtered toimpose the CHIME passband and then passed through a three-way splitter. Two of the threesplitter outputs are connected to coaxial cables which bring the calibration signals to the base ofthe two cylinders. They each terminate in a helix antenna located at the bottom of the CHIMEcylinders, which injects a circularly polarized signal into the CHIME feeds. The third port ofthe splitter is connected directly to the CHIME correlator through a delay cable. The delaycable is required to ensure that the injected signals arrive at the channelizer well within theFFT ( ∼ µ s)length for channelizing into 400 – 800 MHz.The attenuation of the reference channel is adjusted so that its level at the input of the Analogto Digital Converter (ADC) minimizes the noise penalty due to quantization and ADC noise.The attenuation of the BIS signals is adjusted so the additional noise contribution is ∼
50 K,roughly doubling the CHIME system noise temperature. This ensures that the combined signallevel from the sky and BIS remains higher than our quantization noise but low enough thathe amplifiers and ADCs remain in their linear signal regime.
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We are currently working toimplement a gating scheme in the CHIME correlator which will allow us to reduce the time wehave doubled our noise with BIS to short, occasional bursts.
We demonstrate this strategy with a small data set (18 minutes of data with 4 channels fromeach cylinder). We are continuing to improve the system and in particular working to choosethe period and duration of the injected noise, but the preliminary results are encouraging: wecan already calibrate to ∼
1% with the current injection setup.The BIS signal has a Gaussian distribution with zero mean and is broadband, allowing usto calibrate the entire CHIME bandpass simultaneously. The resulting measured visibility is aversion of Equation 2 with gain terms only. The measured visibilities V meas for BIS on and BISoff are: V meason = gg H σ s + GV sky G H + N on (10) V measoff = GV sky G H + N off (11)where V sky is the N × N matrix containing the true sky complex visibility, G is the N × N matrixwith complex receiver gains ( g ) on the diagonal, and σ s is the BIS signal level (the directiondependent array response for the BIS is assumed to be known and is not shown explicitly).The two receiver noise terms N on and N off will not in principle be exactly identical but theirdifference is a white noise term and so decreases as 1 / √ τ ∆ ν as long as both the sky and receiverresponse remains unchanged during observations. With an accurate measurement of σ s fromthe auto-correlation of the noise source channel, the difference between the measurement withBIS on and BIS off, divided by σ s is:∆ V meas σ s = gg H + ∆ N σ s (12)yielding a quantity proportional to the receiver gain with a noise bias. In practice, the data foreach pulse (BIS on, BIS off) is separately averaged and the mean values are differenced. Thisproduces a data set of differenced values at half the cadence of the switching, in this case 10 s (5 son, 5 s off, with a data sample time of 84 ms). The differenced time stream is a noisy estimatorfor the gain solution, which can be obtained by minimizing the residual noise in the differenceddata set. This can be efficiently solved by formulating this as a rank-1 approximation problem(singular value decomposition, SVD), and for each time bin we solve the following to find anestimate of the length- N gain vector ˆg : ˆg = argmin g (cid:13)(cid:13)(cid:13)(cid:13) ∆ V meas σ s − gg H (cid:13)(cid:13)(cid:13)(cid:13) F = (cid:112) λ max u max (13)where λ max is the largest eigenvalue for each channel and u max is its corresponding eigenvectorin the eigenvalue decomposition of ∆ V meas .t can be shown that the error on the estimated gain is determined by the ratio betweenthe first and second largest eigenvalues in Equation 13. We call this the dynamic range, and itshould be < −
20 dB to achieve 1% relative gain calibration precision. The dynamic range forthe BIS measurement is shown in Figure 6, and in these preliminary measurements is ∼ −
20 dBfor most of the CHIME band.Figure 6: The ratio of the first and second eigenvalues in the eigenvalue decomposition of ∆ V (‘dynamic range’) obtained during the first tests of the white spectrum signal injection. Thedynamic range determines the accuracy with which we can recover the receiver gains and is −
20– 25 dB in most of the CHIME band, corresponding to relative gain errors below 1%. Light ∼ We have observed pulsar transits with CHIME alone. These function as a switching noise sourcemuch like BIS. The formalism and gain estimation techniques from Section 4.2 apply to pulsarsas well as the noise source, and can be used to extract relative gains between channels. Using aseries of fast-cadence pulsar data from CHIME over the course of three days, the complex gain(amplitude and phase) stability were assessed. The resulting phase stability, relative to a singlechannel, is shown in Figure 8. The data is also preliminary and cannot yet assess stability fromthis measurement.igure 7: Gain stability (both amplitude and phase) performance of a typical receiver ( ν = 644MHz). Top:
Relative gain amplitude and phase during the 18 minute noise injection data file.
Bottom : Power spectrum of the relative gain amplitude and phase from the time stream. To fitthe power spectrum and derive stability, we will take additional data with with longer files andfaster switching.Figure 8: Relative phase stability averaged across frequencies during a pulsar transit over threedays for three feeds on a single cylinder referenced to the phase of a fourth. This fiducial channel(blue) has been set to − π . The data is preliminary and include variation between pulses whichare intrinsic to the pulsar. The purpose of the data was primarily to understand timing, cabledelays, and other instrumental properties but is still an initial step towards understanding thegain stability using pulsars. .4 Redundant Baselines The large number of redundant baselines within CHIME may be exploited for calibration pur-poses. For an array with N feeds there will be N measured visibilities. If the primary beamsof each feed are identical, the true-sky visibilities V s ij will be common to all measurementswith the same baseline separation and can be written as a simplified version of Equation 2: V meas ij = g i g ∗ j V sky ij , with the same baseline separation u ij . This creates an overdetermined sys-tem that can be solved simultaneously for the complex gain, g i , and the true-sky visibilitiesindependent of any ancillary data up to an overall calibration scale factor and sky tilt. Several different algorithms have been developed to solve redundant baseline systems and the method has been tested on existing arrays.
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We have developed an efficient, hybridmethod for solving the system: to solve for the amplitude of the complex gain, we build on thework of Liu et al and linearize the problem by taking the logarithm of the system. To solve forthe phase, we use an SVD technique applied to the difference, V meas ij − g i g ∗ j V sky ij , to find the bestrank-1 approximation to the gain matrix g i g ∗ j . In simulations of a 12-feed instrument observinga realistic sky signal with forecast instrument noise and identical primary beams, we recoverthe input relative gain amplitude without statistical bias to a precision of better than 0.1%for each 10 s observation and 0.39 MHz frequency band. In the same simulation, we recoverthe gain phase with an rms uncertainty of 0.7 ◦ per observation and frequency band, wherethe requirement is ∼ ◦ ) and is included in the simulations to derive a calibrationrequirement on complex gain.In a real instrument, the primary beam of each feed will differ slightly from the average, whichleads to non-redundant visibilities even if the baselines are redundant. To see how importantsuch effects are for our calibration determination, we produced a simulation in which we perturbthe FWHM of each primary beam by 2% (rms) and solve the redundant baseline system ignoringthe beam variations. In this case, our recovered gain amplitude has a systematic error of 0.2%,which exceeds its 0.1% statistical uncertainty.We are investigating extensions of the above algorithm that can accommodate primary beamvariations and have had some success with their early implementation. Our approach is toexpand the primary beam in a perturbative expansion and to simultaneously solve for the skysignal, the complex gain, and the beam expansion coefficients. To date, we have treated thebeam as a 2-term expansion in Hermite polynomials and have applied this technique to theperturbed beam simulation described above. In that test, we were able to recover the inputgain without statistical bias; the rms uncertainty was ∼
20% larger than the idealized redundantalgorithm, but it was still less than 0.1% per observation and frequency band. We continue todevelop this complementary approach to the calibration of CHIME.
5. CONCLUSION
We have developed requirements for CHIME calibration and formed a plan to calibrate thepolarized beams and complex gains of the CHIME instrument. We have a simulation pipelinecurrently in place that has already provided valuable specifications for the beam and gain calibra-tion requirements. We are working on using it to assess acceptable levels of cross-talk, aliasing,nd instrumental passband, particularly where they may inform instrument design such as feedspacing.We will have made the first pulsar holography measurements to map the polarized beams inMay 2014. We have implemented a basic noise injection system which has proven to calibratecomplex gains to 1%, our target for controlling gains on short time scales. We have also confirmedthat our system noise is on-target. Further progress on improvements to the noise injectionsystem, calibration of the 26 m dish at the DRAO for pulsar holography, and testing redundantbaseline algorithms on data will be done in the near future.
ACKNOWLEDGMENTS
We are very grateful for the warm reception and skillful help we have received from the staffof the Dominion Radio Astrophysical Observatory, operated by the National Research CouncilCanada.We acknowledge support from the Canada Foundation for Innovation, the Natural Sciencesand Engineering Research Council of Canada, the B.C. Knowledge Development Fund, le Co-financement gouvernement du Qu´ebec-FCI, the Ontario Research Fund, the CIfAR Cosmologyand Gravity program, the Canada Research Chairs program, and the National Research Councilof Canada. M. Deng acknowledges a MITACS fellofowship. We thank Xilinx and the XUP fortheir generous donations.
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