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MNRAS 000, 1–16 (2017) Preprint 21 October 2018 Compiled using MNRAS LATEX style file v3.0

Simulated predictions for H i at z = 3.35 with the Ooty

Wide Field Array - I. Instrument and the foregrounds

Visweshwar Ram Marthi1⋆, Suman Chatterjee2,3, Jayaram N Chengalur1,

Somnath Bharadwaj2,31National Centre for Radio Astrophysics, Tata Institute of Fundamental Research, Post Bag 3, Ganeshkhind, Pune - 411 007, India2Department of Physics and Meteorology, Indian Institute of Technology Kharagpur, Kharagpur - 721 302, India3Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur - 721 302, India

Accepted 2017 July 14. Received 2017 June 25; in original form 2016 November 22

ABSTRACT

Foreground removal is the most important step in detecting the large-scale redshiftedH i 21-cm signal. Modelling foreground spectra is challenging and is further compli-cated by the chromatic response of the telescope. We present a multi-frequency angularpower spectrum(MAPS) estimator for use in a survey for redshifted H i 21-cm emissionfrom z ∼ 3.35, and demonstrate its ability to accurately characterize the foregrounds.This survey will be carried out with the two wide-field interferometer modes of theupgraded Ooty Radio Telescope, called the Ooty Wide Field Array(OWFA), at 326.5MHz. We have tailored the two-visibility correlation for OWFA to estimate the MAPSand test it with simulated foregrounds. In the process, we describe a software modelthat encodes the geometry and the details of the telescope, and simulates a realisticmodel for the bright radio sky. This article presents simulations which include the fullchromatic response of the telescope, in addition to the frequency dependence intrinsicto the foregrounds. We find that the visibility correlation MAPS estimator recoversthe input angular power spectrum accurately, and that the instrument response tothe foregrounds dominates the systematic errors in the recovered foreground powerspectra.

Key words: instrumentation: interferometers; methods: data analysis, numeri-cal, statistical; techniques: interferometric; cosmology: observations, diffuse radiation,large-scale structure of Universe

1 INTRODUCTION

Measuring the post-reionisation neutral hydrogen powerspectrum provides a new probe of the large scale structure ofthe universe (Bharadwaj et al. 2001; Wyithe & Loeb 2009;Pritchard & Loeb 2012), and would allow constraining manyimportant cosmological parameters (Wyithe et al. 2008;Loeb & Wyithe 2008; Bharadwaj et al. 2009; Visbal et al.2009; Chang et al. 2010; Seo et al. 2010; Bull et al. 2015;Padmanabhan et al. 2015). The surge of interest in 21-cm intensity mapping over the last decade stems fromthe promise of constraining dark energy. The expansion ofthe universe is thought to accelerate at late times due todark energy. The Baryon Acoustic Oscillation (BAO) hasbeen proposed as a standard ruler (Eisenstein 2005). Itsmeasurement at multiple redshifts can measure the rateof expansion of the universe over cosmic time and hence

⋆ E-mail: vrmarthi@ncra.tifr.res.in

place constraints on dark energy (Wang 2006; Chang et al.2008). More recently, 21-cm intensity mapping has gainedattention due to its potential to constrain the neutrinomasses (e.g. Pritchard & Pierpaoli 2008, 2009; Oyama et al.2013; Villaescusa-Navarro et al. 2015). A number of post-reionisation experiments are currently being planned: in-terferometric experiments, like the Canadian HydrogenIntensity Mapping Experiment (CHIME; Bandura et al.2014) and those with other telescopes such as the OotyWide Field Array (OWFA;Prasad & Subrahmanya 2011;Subrahmanya et al. 2017), the Baryon Acoustic OscillationBroadband and Broad Beam Array (BAOBAB; Pober et al.2013a), and the Tianlai Cylinder Radio Telescope (CRT;Chen 2011), as well as single dish intensity mapping experi-ments like BINGO (Battye et al. 2016) and GBT-HIM (e.g.Chang et al. 2010). The goals of these experiments are sim-ilar - to constrain cosmological parameters through a mea-surement of the power spectrum of the redshifted H i 21-cmsignal and to measure the BAO peak through a measure-

c© 2017 The Authors

2 Marthi et al.

ment of the power spectrum of the density fluctuations. Thisarticle discusses the OWFA effort.

Ali & Bharadwaj (2014) have presented calculations ofboth the H i signal and the foregrounds expected for OWFA,considering the diffuse Galactic synchrotron and the extra-galactic radio source contribution. Bharadwaj et al. (2015)present Fisher-matrix forecasts of the H i signal for OWFA,where they predict that a 5σ detection of the amplitude ofH i signal is achievable in ∼ 150 hours of integration, assum-ing that foregrounds have been completely removed.

The 39-MHz observing band at OWFA, centred at 326.5MHz, provides access to the H i signal from a window of∆z ∼ 0.52 around z ∼ 3.35. A statistical detection ofthe post-reionisation signal is fraught with challenges verysimilar to those in low frequency radio epoch of reionisa-tion (EoR) experiments, the most formidable of which isdealing foreground emission (see e.g. Furlanetto et al. 2006;Ali et al. 2008). Foregrounds are important at least for tworeasons: (1) the astrophysical foregrounds are many ordersof magnitude brighter than the cosmological signal, and (2)foregrounds interact with the instrument to produce spec-tral signatures that can contaminate the H i signal (see e.g.Vedantham et al. 2012; Thyagarajan et al. 2015a).

At least two classes of techniques to treat foregroundsexist. In foreground removal methods, the foregroundsare mathematically modelled as spectrally smooth func-tions and subtracted in the image space (Morales et al.2006; Jelic et al. 2008; Bowman et al. 2009; Liu et al.2009b; Chapman et al. 2012) or from the visibilities(McQuinn et al. 2006; Gleser et al. 2008; Liu et al. 2009a;Petrovic & Oh 2011). Ghosh et al. (2011a,b) fit smoothfunctions to the multi-frequency angular power spectrum ateach multipole to subtract the foregrounds, exploiting thefact that the H i signal decorrelates more rapidly with fre-quency (Bharadwaj & Sethi 2001; Bharadwaj & Ali 2005)than the foregrounds. Foreground isolation, on the otherhand (Datta et al. 2010; Morales et al. 2012; Parsons et al.2012; Vedantham et al. 2012; Thyagarajan et al. 2013;Liu et al. 2014), uses only those regions of the observedk−space that can be identified as being less affected by theforeground emission.

Although it is known that the major foreground com-ponents all have smooth spectral behaviour, allowing themto be distinguished from the cosmological 21-cm signal, thetelescope’s chromatic response introduces spectral featuresin the foreground contribution to the observed signal. Theseeffects, the details of which are telescope-specific, pose a se-rious challenge for any foreground removal technique. Sim-ulations that incorporate a detailed frequency dependenttelescope model are crucial to quantifying these effects aswell validating any foreground removal technique. In thispaper we present a software emulator that describes a chro-matic model for OWFA. The emulator is used in conjunc-tion with a foreground sky model to predict the foregroundsexpected in observations with OWFA. The three dimen-sional power spectrum P (k) of the redshifted 21-cm bright-ness temperature fluctuations (e.g. Bharadwaj & Ali 2005)is well suited to quantifying the H i signal which, to a goodapproximation, may be assumed statistically homogeneousin the three dimensional space spanned by the two angu-lar coordinates and the frequency axis. This assumption,however, is not valid for the foregrounds, where the statis-

tics are quite different in the angular and frequency direc-tions. The issue is further complicated by the telescope’schromatic response. The Fourier modes k are no longer theideal basis, and the power spectrum P (k) is no longer theobvious choice to quantify the statistical properties of themeasured sky signal. In this paper we propose using the cor-relations between the visibilities to quantify the statisticalproperties of the measured sky signal. The relation betweenthe visibility correlations and the power spectrum P (k) hasbeen studied in several earlier works (Bharadwaj & Sethi2001; Bharadwaj & Ali 2005; Ali & Bharadwaj 2014) and iswell understood. The multi-frequency angular power spec-trum (MAPS; Datta, Choudhury & Bharadwaj 2007) pro-vides a technique to jointly characterize the angular and fre-quency dependence of the observed sky signal. This is veryclosely related to the visibility correlations (Ali et al. 2008;Ali & Bharadwaj 2014), and we also use this to quantifyour simulated foreground predictions for OWFA. In whatfollows we distinguish between three different kinds of visi-bilities, viz.: (a) the model visibilities, which are the Fouriertransform of the primary beam-weighted sky brightness dis-tribution for the array geometry under consideration, (b) themeasured visibilities, which are the actual measurements ofthe beam-weighted sky made by the array, corrupted by thecomplex antenna gains and (c) the true visibilities, which areobtained from the measured visibilities after the process ofcalibration, which is a noisy version of the model visibilities.

To begin with, we introduce OWFA, the instrument, inSection 2, and describe the emulator in Section 3. In Sec-tion 4 we introduce the foreground components and describethe steps in the simulation of the foreground fields. Themodel visibilities are obtained for the simulated foregrounds,given the instrument description. Section 5 introduces themulti-frequency angular power spectrum (MAPS) estima-tor and its construction from the visibility correlation forOWFA, which we use to estimate the angular power spec-trum from the true visibilities. Section 6 presents the resultsof the simulation and power spectrum estimation. We followthem up with a discussion of the properties of the estimatorin Section 7.

2 THE OOTY WIDE FIELD ARRAY - OWFA

The Ooty Radio Telescope (ORT; Swarup et al. 1971) is lo-cated in the Nilgiri Hills in the Indian peninsula, approxi-mately at 11N at an altitude of ∼ 8, 000 ft. The telescope,operating at 326.5MHz, has an offset-parabolic cylindricalreflector whose aperture is a 530m long and 30m wide rect-angle as shown in Figure 2. The feed consists of 1056 dipoles,each 0.5λ long, all arranged end to end along the length ofthe line focus of the reflector. The telescope is located on ahill that is roughly sloped 11 North-South, which equals thelatitude of the station. Figure 1 shows a part of the ORT.Effectively, the axis of the cylinder is parallel to the earth’srotation axis, making it equatorially mounted. The telescopeis steerable in the East-West direction mechanically by ro-tating the cylinder about its axis, allowing continuous track-ing of a source on the sky from rise to set. The telescope’sresponse is steered electronically in the North-South direc-tion by switching the phases of the dipoles through an ana-log phase-switching network. Here we consider observations

MNRAS 000, 1–16 (2017)

Foreground predictions for OWFA 3

Figure 1. (colour online) A photograph of the Ooty Radio Tele-scope (ORT) on the sloping hill, showing four of the 22 parabolicframes. The left side of the picture is North and the right is South.

pointed towards the phase center (α, δ) = (α0, δ0), denotedby the unit vector m (Figure 2).

In the legacy ORT system the signals from the dipolesare combined in a passive combiner network tree, only a partof which is shown Figure 3. The signals from 24 successivedipoles are combined hierarchically in a series of 4-way, 2-way and 3-way combiners. The output at this stage is calleda half-module. Two half-module outputs are again combinedin a 2-way combiner: therefore the 1056 dipoles are groupedinto 22 modules of 48 dipoles each. The 22 modules aresplit into 11 Northern and 11 Southern modules, each halfis used to generate 11 phase-shifted beams, each pointing ata different direction, and 1 total intensity beam. The beamsobtained from the Northern and Southern modules are cor-related with each other to get rid of systematics. The legacybeam-former mode of the ORT system provides twelve si-multaneous beams on the sky.

The ORT is presently undergoing a major up-grade to operate as a programmable interferometer(Subrahmanya et al. 2017). The upgrade will enable twoconcurrent but independent interferometer modes, besidesretaining the workhorse beam-former mode. The two con-current modes are:

(i) P-II : the signal is tapped after the 4-way combiner -the second stage of the combiner network (Figure 3) - to givea field-of-view (FoV) of 1.8×27 EW×NS and a bandwidthof ∼ 39MHz. It would operate as a linear array of 264 equi-spaced antennas, giving ∼ 35, 000 baselines.

(ii) P-I: every group of six adjacent 4-way combiner out-puts are summed in phase in the software correlator system.This is equivalent to the signal being tapped at every half-module(24 dipoles), giving a FoV of 1.8×4.5 EW×NS, and∼ 39MHz bandwidth. It would operate as an independentlinear array of 40 equi-spaced antennas, giving 780 baselines.

The RF signals tapped at the 4-way combiner (Figure 3)are digitised in the field and the signals are transported tothe central building on optical fibre. The signals undergo twolevels of pooling and a protocol conversion before arrivingin a high-performance cluster (HPC) that will correlate the

WE

S

N

530 m

30 m

PSfrag replacements

x

yz

m

n

Figure 2. This shows a diagram of the 30m × 530m aperture ofthe ORT parabolic cylinder whose axis is alighned with the N-Sdirection. The smaller rectangles, one of which is shaded, depcitthe apertures of the individual OWFA antennas which are linearlyarranged along the length of the ORT cylinder. The unit vectorn denotes an arbitrary direction with coordinates (α, δ) on thecelestial sphere. Our Cartesian coordinate system which is tied tothe telescope has the z-axis in the N-S direction, and the x-axisalong the normal to the aperture pointing towards the position(α0, 0) on the celestial equator. The unit vector m denotes the

phase center (α0, δ0) which can be steered electronically in theN-S direction.

signals to obtain the visibilities. The P-I system mentionedabove will run in parallel with the P-II system within theHPC.

OWFA will operate as an array of NA antennas ar-ranged linearly along the length of the 530m ORT cylin-der as shown in Figure 2. Each OWFA antenna has a b× drectangular aperture, where b = 30m is the width of theparabolic reflector and d is the length of the antenna whichis different for P-I and P-II. d is also the spacing betweenan adjacent antenna pair. The array parameters are sum-marized in Table 1.

The regular spacing of the antennas results in a largefraction of the NAC2

1 baselines being redundant, and thereare only NA − 1 unique baselines. We can express theseunique baselines in terms of the smallest baselineU 1 = dν/cas

U n = nU 1 with 1 6 n 6 NA − 1 . (1)

Each baseline U n has a redundancy factor of NA − n.OWFA is conveniently described using a right-handed

Cartesian coordinate system tied to the telescope as shownin Figure 2. The z-axis is along the N-S direction whichis parallel to the axis of the parabolic cylinder, and the x-axis is normal to the telescope’s aperture. In this coordinatesystem the baseline vectors are all along the z-axis and wehave

U n = |Un|z . (2)

1 nCr is defined as n!(n−r)!r!

MNRAS 000, 1–16 (2017)

4 Marthi et al.

21 2322 245 6 7 8

Amplifier +

directional coupler

Amplifier +

directional coupler

Amplifier +

directional coupler

Amplifier +

directional coupler

In−fielddigitiser

In−fielddigitiser

In−fielddigitiser

In−fielddigitiser

Legacysystem

Legacysystem

. . .1 2 3 4 20191817

4−way combiner 4−way combiner 4−way combiner 4−way combiner

2−way combiner2−way combiner

Figure 3. The signal paths from the dipoles to the digitiser of OWFA: each dipole has an LNA and a phase shifter for setting thedeclination. The output of the 2-way combiner proceeds to the legacy ORT receiver system. The legacy ORT receiver system is abeam-former network capable of producing twelve simultaneous beams on the sky.

The unit vector n towards an arbitrary direction (α, δ) onthe celestial sphere can be expressed as

n = sin(δ) z + cos(δ)[cos(α− α0) x + sin(α− α0)y ] . (3)

where x , y and z respectively denote the unit vectors alongthe x-, y- and z-axes. The unit vector m towards the phasecenter (α0, δ0) can be represented as

m = sin(δo) z + cos(δ0)x . (4)

Observations at 326.5 MHz relate to H i at z = 3.35,which is at a comoving distance of r = 6.84Gpc. This re-lates each baseline U to k⊥ = 2π|U |

rwhich is the compo-

nent of a 3D wave vector k perpendicular to m . This setsthe range kmin

⊥ and kmax⊥ that will be probed by OWFA

(Ali & Bharadwaj 2014). Since OWFA is one-dimensional,it only probes k⊥ along the length of the interferometer.We also have r′ = |dr/dν| = 11.5MpcMHz−1 which setsthe comoving interval L = r′ Bbw spanned by Bbw the sys-tem bandwidth of OWFA. This sets kmin

‖ = 2πr′ Bbw

and

kmax‖ = π

r′ (∆ν)where ∆ν is the channel width (frequency

resolution) and k‖ is the component of k parallel to m .Table 1 summarizes the important parameters of the twointerferometer modes, assuming the WMAP 9-year resultsfor the cosmological parameters (Hinshaw et al. 2013).

3 A SOFTWARE EMULATOR FOR OWFA

In this Section, we will describe a software emulator forOWFA that we subsequently use to make foreground pre-dictions. The emulator (Prowess; Marthi 2017) was written(1) to simulate the observed radio sky at 327 MHz, (2) toprovide a pipeline to simulate visibilities, introduce extrin-sic systematics, study intrinsic systematics and allow us totest calibration schemes (see Marthi & Chengalur 2014) and

Table 1. Parameters of OWFA. OWFA is a one-dimensional in-terferometer; the values for k pertain only to the north-southextent of the array elements.

Parameter P-I P-II

Antennas 40 264Total baselines 780 34716Unique baselines 39 263Shortest baseline 11.5 m 1.92 mLongest baseline 448.5 m 505.0 mCentral frequency 326.5 MHz 326.5 MHzBandwidth 39 MHz 39 MHz

Spectral resolution 125 kHz 125 kHzAperture (b × d) 30× 11.5 m2 30× 1.97 m2

FoV at δ = 0 1.8 × 4.5 1.8 × 27

Resolution 0.1 sec δ0 × 1.8 0.1 sec δ0 × 1.8

kmin⊥ 1.1× 10−2 Mpc−1 2.0× 10−3 Mpc−1

kmax⊥ 4.6× 10−1 Mpc−1 5.2× 10−1 Mpc−1

kmin‖

1.4× 10−2 Mpc−1 1.4× 10−2 Mpc−1

kmax‖

4.6 Mpc−1 4.6 Mpc−1

detection strategies even as the telescope undergoes an up-grade, (3) to prepare a science-ready observatory data anal-ysis software pipeline in time for first light. For the purposeof this article, only the first and second items are relevant.We describe Prowess2 and some of the details of its de-sign briefly in this section. It is important to note that thetelescope model incorporated in the emulator includes thechromatic effects associated with the primary beam and thebaselines.

2 Prowess is available at https://github.com/vrmarthi/prowess

MNRAS 000, 1–16 (2017)

Foreground predictions for OWFA 5

d d d d d dd

b

z

z

d d d d d d

(a) Aperture arrangement (b) Primary beam

d/λ 2d/ λ 3d/λ

A(U, ν)∼

0 u

2λ /bd

(c) Baseline space

Figure 4. (colour online) The aperture arrangement for OWFA in real and baseline space.

3.1 Visibilities

The visibilities M (U ) are the Fourier transform of the prod-uct of the specific intensity distribution on the sky I(n , ν)and the telescope’s primary beam pattern A(∆n , ν) where∆n = n − m (Figure 2). The visibilities can be calculatedas

M (U ) =

∫

dΩn I(n , ν)A(∆n , ν) e−2πiU ·∆n (5)

where the solid angle integral is over the entire celestialhemisphere.

Here A(∆n , ν) is the primary beam power pattern ofOWFA, modelled as a two-dimensional sinc-squared func-tion

A(∆n , ν) = sinc2(

πbν∆ny

c

)

sinc2(

πdν∆nz

c

)

(6)

where ∆ny and ∆nz are respectively the y and z compo-nents of ∆n . The primary beam pattern here is taken tobe the diffraction pattern of rectangular aperture of dimen-sions b × d, the values of b and d for OWFA are given inTable 1. This form of the primary beam has been assumedas it likely to lead to larger systematics because of its highsidelobe levels and is hence conservative. In reality, the ta-pered east-west illumination of the aperture results in anapproximately truncated Gaussian beam. Prowess is capa-ble of handling any user-defined primary beam. Figure 4(a)shows the aperture arrangement and Figure 4(b) shows theP-I primary beam pattern as a function of the celestial co-ordinates α and δ for a pointing towards (α0, δ0) = (0, 0).

Note that in this co-ordinate system the phase factorwhich is the argument of the exponent in equation (5)

2π iU ·∆n = 2π i |U |[sin(δ)− sin(δ0)] (7)

depends only on the baseline length and the declinations,reflecting the 1D geometry of OWFA.

3.2 Overview of the software pipeline

We begin with a geometric description of the array in aninput Antenna Definition file. Since the telescope is equato-rially mounted, the baseline co-ordinates remain stationaryas the antennas track the pointing coordinates on the sky.The baseline vectors are obtained from the physical antenna

separations defined at the central frequency but scaled ap-propriately at each channel when computing the visibilities.

d |a−b| = x a − x b, U |a−b| = d |a−b|ν

c(8)

It is worth noting that the chromatic response of the inter-ferometer enters the picture through equations (6) and (8)which respectively capture the fact that the primary beampattern and the baseline distribution both vary with fre-quency. Even for a perfectly achromatic source, the chro-matic response of the interferometer will introduce a fre-quency dependent structure in the measured visibilities.

The pixel sizes in all our simulated maps and the pri-mary beam are ∼ 1′ × 1′. The maps and the primary beamare all 2048× 2048 pixels wide, spanning an angular extentof ∼ 38 in each dimension. In this paper, we will focus onlyon P-I, where the antenna primary beam full width at halfmaximum (FWHM) is 1.8 × 4.5.

The brightness distribution for the diffuse foregroundcomponent in our simulations is obtained as a random re-alisation of its power spectrum (see Section 4). The specificintensities I(np, ν) of the pixelized sky map are convertedto flux densities by scaling with the appropriate solid angle

S(np, ν) = I(np, ν)∆Ω , (9)

the index p here refers to the NP pixels in the map. The ex-tragalactic point source foreground is obtained as a randomrealisation of its power spectrum and obeying the differen-tial source count relation (see Section 4). This results in aset of NS discrete extragalactic radio sources identified bytheir respective positions ns and flux densities S(ns, ν).

We note that similar simulations of both the instru-ment and foregrounds have been carried out for CHIME(Shaw et al. 2014) and MWA (Thyagarajan et al. 2015a).In contrast to the above full sky foreground simulations weemphasize that for P-I of OWFA, where the primary beam ismuch smaller in comparison, a full sky treatment of the fore-grounds is not necessary. However, unlike the small FoV LO-FAR foreground simulations (Jelic et al. 2008), we do con-sider a FoV much further out than the primary. For P-II,with a primary FoV of 0.5 rad, wide-field effects become im-portant, and they will be considered separately for the P-IIforecasts.

The model visibilities M (U ) for the non-redundant set

MNRAS 000, 1–16 (2017)

6 Marthi et al.

of baselines are obtained using a discretised version of equa-tion (5):

M (U ) =∑

b

S(nb, ν) A(∆nb, ν) e−i2πU ·∆nb . (10)

The sum denoted here by the index b includes the NP pixelsof the diffuse foreground as well as the NS discrete extra-galactic sources.

The visibility V measured by a baseline with antennasa and b depends on the model visibility for that particularspacing, as well the element gains:

V (U ab) = ga g∗b M(

U |a−b|

)

+N (U ab) (11)

where ga and gb are the complex antenna gains (which couldalso be frequency dependent) and N is Gaussian randomnoise equivalent to the system temperature Tsys.

The antenna gains and the true (calibrated) visibili-ties are estimated from the measured visibilities by employ-ing the non-linear least squares minimisation algorithm de-scribed in Marthi & Chengalur (2014). Due to the enormousredundancy of the measurements, the highly overdeterminedsystem of equations enables simultaneously solving for boththe gains and the true visibilities. This is interesting particu-larly in light of studies regarding the limitations introducedby calibration based on an incomplete model of the sky, asis routinely employed in interferometry Patil et al. (2016).The real and imaginary part of the noise N in equation (11)each has a RMS fluctuation (Thompson et al. 2008)

σab =

√2kBTsys

ηA√∆ν∆t

(12)

per channel, where kB is the Boltzmann constant, η is theaperture efficiency, A = b × d is the aperture area, ∆ν thechannel width and ∆t the integration time. The foregroundmaps give the flux in Jy units at every pixel, therefore theFourier sum directly produces the model visibilities in Jyunits. The flowchart in Figure 7 of Marthi (2017) gives anoverview of the part of the emulator used to obtain the vis-ibilities. As mentioned earlier, the emulator fully incorpo-rates the chromatic response of the instrument. The base-lines (eq. 8), the primary beam (eq. 6) and the sky varywith frequency. These quantities were calculated separatelyat each frequency and used in equation (10) to calculate thevisibilities. The spectral variation of the sky is discussed inSection 4. The pixel solid angle ∆Ω in equation ( 9) is heldfixed across the band. In our simulations of the foregrounds,the sky is sampled at a much higher resolution than theresolution element of OWFA. The simulated visibilities, theprimary beam and the foreground maps are written to diskin the Flexible Image Transport System (FITS; Wells et al.1981) format. The visibility records are written to UVFITSfiles at one-second interval.

In Prowess, the observing band is centred at 326.5MHz with a bandwidth of ∼ 39 MHz split into 312 chan-nels in the simulations. The frequency resolution is thus125 kHz per channel. Based on our understanding of thedistribution of the neutral gas around the redshift of z ∼3.35, the H i signal at two redshifted frequencies, sepa-rated by more than ∆ν ∼ 1 MHz, is expected to decorre-late rapidly (Bharadwaj & Ali 2005; Bharadwaj et al. 2009;Chatterjee et al. 2017). This means that with a channel res-olution of 125 kHz, the H i signal correlation is adequately

sampled over the 1-MHz correlation interval. In reality, thechannel resolution is likely to be much finer (∼ 50kHz), withabout 800 channels across the 39-MHz band. This is usefulfor the identification and excision of narrow line radio fre-quency interference. Beyond the need to handle RFI, thereis no incentive to retaining the visibilities at this resolution.Eventually, we may smooth the data to a resolution of 125kHz, keeping in mind the decorrelation bandwidth of the H i

signal. Prowess itself is indeed capable of running at anyfrequency resolution, including the actual final configurationof the P-I and P-II systems. But the 125-kHz resolution usedin the simulations allows for rapid processing, especially ifthey are to be run repetitively for a wide range of differentparameters.

4 THE FOREGROUNDS

The most dominant contributors to the foregrounds arethe diffuse synchrotron emission from our Galaxy and theextragalactic radio sources (see e.g. Di Matteo et al. 2002;Santos et al. 2005; Furlanetto et al. 2006; Ali et al. 2008).Although some choice is available in the selection of a rea-sonably cold region of the Galaxy through which to ob-serve the distant universe, emission from extragalactic radiosources is ubiquitous. At different angular scales, either ofthe two dominates: at the very large scales, ℓ . 100, theGalactic synchrotron emission contributes more to the fore-ground budget. At smaller scales, there is a gradual tran-sition to the flux from extragalactic sources. Galactic syn-chrotron is likely to dominate the emission at the largestangular scales observable by P-II at the OWFA, and likelyat the first few baselines of P-I (Ali & Bharadwaj 2014) athigher declinations. Since the largest contribution to the H i

signal is expected to come from the largest angular scales(Chatterjee et al. 2017) we would have access to, the largescale emission from the diffuse foreground assumes impor-tance.

4.1 The Diffuse Galactic Foreground

Diffuse synchrotron emission from the Galaxy has been stud-ied extensively, both within the context of statistical H i ex-periments (Jelic et al. 2008; Bernardi et al. 2009; Jelic et al.2014) as well as in their own right (e.g. Haslam et al.1981, 1982). La Porta et al. (2008) determine the powerspectrum of the diffuse synchrotron emission at scaleslarger than 0.5o from the 408-MHz Haslam total inten-sity maps, where they find that the index of the powerlaw γ varies between 2.6 and 3.0. Similarly, there are mea-surements at scales smaller than 0.5o from the WSRT(Bernardi et al. 2009, 2010), GMRT(Ghosh et al. 2012) andLOFAR (Iacobelli et al. 2013).

We assume that the fluctuations in the diffuse Galac-tic Synchrotron radiation are statistically homogeneous andisotropic, described by a Gaussian random field whose sta-tistical properties are completely specified by the angularpower spectrum. We further assume that the input modelangular power spectrum CM

ℓ (ν) is well described by a singlepower law in the entire range of angular scales of our interest

MNRAS 000, 1–16 (2017)

Foreground predictions for OWFA 7

(a) Simulated map (b) Angular Power Spectrum (c) Angular Power Spectrum withprimary beam

Figure 5. (colour online) (a) shows a particular realization of the simulated brightness temperature map δT (θ, ν) in (α, δ) coordinatesfor the nominal frequency of ν = 326.5MHz.In (b) the black points show the angular power spectrum recovered from the simulated mapshown in (a), the input model angular power spectrum CM

ℓ (ν), is shown by the curve in red. The shaded region shows the multipolerange ℓmin− ℓmax accessible to OWFA. (c) shows the angular power spectrum recovered by applying equation (16) to the image obtainedfor a single realisation by multiplying the simulated map (a) with the primary beam patter shown in Figure 4(b).

(e.g. Santos et al. 2005; Ali et al. 2008):

CMℓ (ν) = Aν0

(ν0ν

)2α(

1000

ℓ

)γ

(13)

where Aν0 is a constant - the amplitude that can be de-termined observationally at ν0 (here at ℓ = 1000), α is thespectral index and γ is the index of the spatial power.

A map of the diffuse synchrotron radiation can be ob-tained as a realization of a Gaussian random field whosepower is distributed in the angular scales described byeqn (13). A detailed description of how to generate sucha realization can be found in Choudhuri et al. (2014), butwe give a brief summary here for completeness. Considerthe model angular power spectrum given in equation (13).We take A150 = 513 mK2, which is the amplitude of theangular power spectrum measured at 150 MHz with theGMRT (Ghosh et al. 2012). For the other paramters, wetake α = 2.52 (Rogers & Bowman 2008) and γ = 2.34(Ghosh et al. 2012). The spectral index α scales the ob-served brightness temperature from a 150-MHz observationto 326.5 MHz as well as imparts an in-band spectral shapeto the diffuse emission.

The foreground simulations have been carried out un-der the flat sky approximation with n = m + θ, where θ

is a 2D vector on the plane of the sky. We use U to de-note the Fourier modes corresponding to θ, and these arerelated to the angular multipole ℓ as ℓ = 2π|U|. The Fouriercomponents of the brightness temperature fluctuations aregenerated on a grid through

∆T (U , ν) =

√

ΩCMℓ (ν)

2(x+ iy) , (14)

where Ω is the total solid angle of the simulated field, andx and y are independent Gaussian random variables withzero mean and unit variance. The map of the brightness

temperature fluctuations δT (θ, ν), or equivalently the spe-cific intensity fluctuations δI(n , ν), is obtained through aFourier inversion. Figure 5(a) shows a single realization ofthe simulated brightness temperature map in δT (θ, ν) in(α, δ) coordinates, for which the recovered angular powerspectrum Cℓ(ν) is shown in Figure 5(b) in comparison withthe input model CM

ℓ (ν) used to obtain the random real-ization. The simulated map is multiplied with the OWFAprimary beam pattern (Figure 4(b)) to obtain δTB(θ, ν) =δT (θ, ν) A(θ, ν) which is used to calculate the model visi-bilities (equation 10). Here ∆TB(U ), which is the Fouriertransform of δTB(θ, ν), is related to the angular power spec-trum Cℓ(ν) as

〈|∆TB(U )|2〉 =∫

d2U ′|A(U −U′)|2C

ℓ′ (ν) (15)

where ℓ′

= 2π|U ′| and A(U ) is the Fourier transform of theprimary beam A(θ, ν). Under the assumption that Cℓ(ν)does not change much within the width of the function|A(U −U ′)|2, the relation between ∆TB(U ) and the angu-lar power spectrum Cℓ(ν) can be (Ali & Bharadwaj 2014)approximated as

〈|∆TB(U )|2〉 =[∫

d2U ′|A(U ′)|2]

Cℓ(ν) (16)

which allows us also to estimate Cℓ(ν) from ∆TB(U , ν) or,equivalently, from the model visibilities M (U ) . Figure 5(c)shows the comparison between the angular power spectrumCℓ(ν) estimated using equation (16) and the input modelCM

ℓ (ν). The recovered angular power spectrum is in goodagreement with the input model for ℓ > 100, whereas theconvolution with the primary beam (equation 15) is impor-tant at smaller ℓ. The fact that |M (U )|2 directly gives anestimate of Cℓ(ν) for most of the ℓ range shown in Fig-ure 5(c) means that the correlation between the visibilities

MNRAS 000, 1–16 (2017)

8 Marthi et al.

(a) Simulated map (b) Angular Power Spectrum (c) Angular Power Spectrum withprimary beam

Figure 6. (colour online) (a) shows a zoomed in view of a single realization of the simulated brightness temperature map δT (θ, ν) in(α, δ) coordinates for the nominal frequency of ν = 326.5MHz. In (b) the black points show the angular power spectrum recovered fromthe simulated map shown in (a), the input model angular power spectrum CM

ℓ (ν), is shown by the curve in red. The shaded region showsthe multipole range ℓmin − ℓmax accessible to OWFA. (c) shows the angular power spectrum recovered by applying equation (16) to theimage obtained for a single realisation by multiplying the simulated map (a) with the primary beam pattern shown in Figure 4(b).Theapproximation is valid throughout the entire ℓ range, except at the smallest multipoles.

measured at OWFA can be used directly to estimate theangular power spectrum of the sky signal, an issue that weexplore in detail later.

The values of the three parameters α, A150 and γ havebeen held constant in our simulations. In reality the spec-tral index α can vary along different lines of sight, and theamplitude A150 and γ is the index of the spatial power canhave different values in different patches of the sky (e.g.La Porta et al. 2008). These variations will introduce addi-tional angular and frequency structures as compared to thepredictions of our simulations.

4.2 Extragalactic radio sources

The dominant emission from the 326.5-MHz sky at most ofthe angular scales of our interest is expected to be fromthe extragalactc point sources (Ali & Bharadwaj 2014).Wieringa (1991) has measured the source counts at 325MHz with the WSRT down to ∼ 4 mJy. More recently,Sirothia et al. (2009) have measured the source counts inthe ELAIS N1 field at 325 MHz with the GMRT, the deep-est till date. Ali & Bharadwaj (2014) have estimated that atmost of the angular scales of P-II and at all scales of P-I, theextragalactic point source contribution is likely to dominatewhen observing at high galactic latitudes. The extragalac-tic radio source power spectrum has a flat Poisson part anda clustered part with a non-zero slope (Cress et al. 1996;Condon 2007; Owen & Morrison 2008; Vernstrom 2015).

Simulating extragalactic radio sources, most of whichare unresolved at the resolution of OWFA, is more involvedthan the straightforward Fourier inversion of a power spec-trum of the diffuse foreground. Details of the method usedhere can be found in Gonzalez-Nuevo et al. (2005) but wesummarize it here for completeness. Let n be the mean num-

ber density of sources per pixel. Note that we have n > 1 fora confusion limited map. We begin with a random Poissonfield of mean source density n, for which the density contrastis defined as δ(θ) = [n(θ)−n]/n, and its Fourier transform isdenoted using ∆(U ). Let its angular power spectrum of δ(θ)be denoted by CP

ℓ , which can be calculated from ∆(U ). Wemodify CP

ℓ by the clustering angular power spectrum Cclℓ :

accordingly the Fourier transform of the density contrast ismodified as

∆′(U ) = ∆(U )

√

CPℓ + Ccl

ℓ

CPℓ

(17)

The modified density contrast spectrum ∆′(U ) is reversetransformed to give the modified density contrast functionδ′(θ). Finally, the modified pixel source density is givenby n′(θ) = n[1 + δ′(θ)]. The input power spectra for thePoisson part and the clustered part are respectively(fromAli & Bharadwaj 2014),

CPℓ =

(

∂B

∂T

)−2 ∫ Sc

0

S2 dN

dSdS (18)

and

Cclℓ = 3.3×10−5 ·

(

∂B

∂T

)−2 [∫ Sc

0

S2 dN

dSdS

](

1000

ℓ

)γ

(19)

The integrals have been computed from a polynomial fitto the source counts from Wieringa (1991). The simulatedmap represents the true source distribution in the sky as nobright sources are assumed to have been subtracted, equiv-alent to an infinite cutoff flux. However, we impose a cutoffof 30 Jy in the numerical integration: this is a reasonablygood approximation to an infinite cutoff flux, as the num-ber of sources beyond ∼ 3 Jy falls very sharply, and thisbehaviour is well captured by the polynomial approxima-tion to the differential source counts. The clustered part

MNRAS 000, 1–16 (2017)

Foreground predictions for OWFA 9

has a power law index of γ = 0.9 (Cress et al. 1996). Themean values are CP

1000 = 1580 mK2 and Ccl1000 = 444 mK2.

We now populate the pixels with radio sources accordingto the observed differential source count relation, drawingfrom the distribution described by the polynomial obtainedfor the fit. The spectral indices for the point source fluxeshave been assigned randomly, drawing from a Gaussian fitto the distribution centered at α = 0.7 with a dispersionof σα = 0.25 (Sirothia et al. 2009) and their channel-wisefluxes scaled accordingly. The angular resolution of thesesimulated maps (∼ 1′ × 1′) is equal to the 325-MHz WSRTresolution of Wieringa (1991), much finer than the OWFAresolution of 6′. Figure 6(a) shows a single realization of thesimulated brightness temperature map in δT (θ, ν) in (α, δ)coordinates. The angular power spectrum of the single ran-dom realization Cℓ(ν), shown in Figure 6(b), is seen to bein good agreement with the input model CM

ℓ (ν) except atthe smallest multipoles. Figure 6(c) shows the comparisonbetween the angular power spectrum Cℓ(ν) estimated usingequation (16) and the input model CM

ℓ (ν).The confusion limit for OWFA is 175 mJy

(Ali & Bharadwaj 2014); however, we do not proposeto identify and subtract sources from the one-dimensionalimages made using the OWFA itself. Instead, a modelfor the sky obtained from deep GMRT imaging could,for example, be used to subtract the contribution to thevisibility from the extragalactic radio sources. Alternatively,the model visibilties upto 18 mJy (the 5σ lmit) could just aswell be obtained from the shallower Westerbork NorthernSky Survey (WENSS; Rengelink et al. 1997).

5 POWER SPECTRUM ESTIMATION

5.1 The two-visibility correlation as the MAPS

estimator

The issue here is to use the measured OWFA visibilitiesV(U n) to estimate the statistics of the background sky sig-nal. Before discussing the method of analysis, it is importantto note that the baseline U n corresponding to a fixed an-tenna separation dn scales as U n = ν dn/c (equation 8)with ν varying across the observing band. This renders itdifficult to use U n and νi as independent parameters to la-bel the visibilities V(U n) that will be measured at OWFA.For the purpose of analyzing the data we adopt the notationwhereU n = νc dn/c which does not vary with frequency butis fixed at νc = 326.5MHz. In this notation we may inter-pret U n and νi as independent parameters that label thevisibilities V(U n, νi).

The proposed method of analysis is here applied to sim-ulated data which only contains the foreground components.As described in the previous section, the foregrounds all havesimple power law frequency dependence in the models whichwe have implemented here. The telescope’s chromatic re-sponse (equations 6 and 8) , however, introduces additionalfrequency dependence which we have attempted to capturein the subsequent analysis.

The two point statistics or, equivalently the power spec-trum of the sky signal, is entirely contained in the two visi-bility correlation

S2(U n, νi;Um, νj) ≡ 〈V(U n, νi)V∗(Um, νj)〉 . (20)

where the angular brackets denote an ensemble average overdifferent random realisations of the sky signal.

In addition to the self-correlation (n = m) , the signalin the adjacent baseline pairs (n = m±1) are correlated (seeFigure 4(c) and Ali & Bharadwaj 2014). Further, the cor-relation is approximately a factor of four smaller comparedto the self correlation if we consider two adjacent baselines(Bharadwaj et al. 2015). For the present analysis we con-sider only the self-correlation, and we use the notation

S2(U n, νi, , νj) ≡ 〈V(U n, νi)V∗(U n, νj)〉 . (21)

The relation between S2 and the power spectrum P (k)of H i is well understood (Bharadwaj & Sethi 2001;Bharadwaj & Ali 2005; Ali & Bharadwaj 2014) and we donot consider it here.

The multi-frequency angular power spectrum Cℓ(νi, νj)(MAPS; Datta et al. 2007) which is defined through

〈∆T (U , νi)∆TB(U′

, νj)〉 = δ2D(U −U′

)Cℓ(νi, νj) (22)

provides an useful tool to jointly characterize the angularand frequency dependence of the statistical properties of thesky signal. Here δ2D(U −U

′

) is the 2D Dirac delta functionand we have ℓ = 2π|U| in the flat sky approximation.

The model visibilities in equation 5 are related to thebrightness temperature fluctuations ∆T (U , ν) as

M (U n, ν) =

(

∂B

∂T

)

ν

∫

d2U A(U n(ν/νc)−U , ν)∆T (U , ν)

(23)

where(

∂B∂T

)

νis the conversion factor from brightness tem-

perature to specific intensity, and we have introduced a fac-tor (ν/νc) in the argument of A to account for the fact thatU n is defined at νc. For the present analysis we assume thatthe visibilities are perfectly calibrated and altogether ignorethe noise contribution. This leads to

S2(U n, νi, , νj) =

(

∂B

∂T

)

νi

(

∂B

∂T

)

νj

∫

d2U × (24)

A(U n(νi/νc)−U , νi)A∗(U n(νj/νc)−U , νj)C2πU (νi, νj)

which relates the visibility correlation to the multi-frequencyangular power spectrum (MAPS). Note that it is possibleto estimate the visibility correlation S2(U n, νi, , νj) directlyfrom the measured visibilities, whereas Cℓ(νi, νj) quantifiesthe intrinsic sky signal independent of any telescope. Equa-tion (24) allows us to relate the statistics of the measuredvisibilities to the intrinsic statistics of the sky signal. We seethat the telescope appears through a convolution in equa-tion (24). It is, in principle, possible to determine Cℓ(νi, νj)from the measured S2(U n, νi, , νj) by deconvolving the effectof the telescope’s primary beam A, and we interchangeablyrefer to S2(U n, νi, , νj) as the visibility correlation or theMAPS estimator.

We now consider the restricted situation where νi =νj , and discuss how it is possible to use the measuredS2(U n, νi, νj) to estimate Cℓ(ν) ≡ Cℓ(νi, νi). We have al-ready seen in equation (14) that for a large part of the ℓrange probed by OWFA it is possible to approximate theconvolution in equation (24) by a multiplicative factor . Thispermits us to approximate S2(U n, νi, νj) as

S2(U n, ν, ν) = KνCℓn(ν) (25)

MNRAS 000, 1–16 (2017)

10 Marthi et al.

where ℓn = 2π|U n| and Kν is defined as

Kν =

(

∂B

∂T

)2

ν

∫

d2U ′|A(U′)|2 , (26)

which is uniquely determined for a known primary beam. Wecan use equation (25) to estimate Cℓ(ν) from the measuredvisibility correlation.

Considering a fixed baseline U n, the visibility correla-tion S2(U n, νi, νj) is (equation 24) a real valued symmet-ric matrix of dimensions Nc × Nc where Nc is the num-ber of frequency channels. Observations will yield such amatrix for every available unique baseline U n. For OWFA(Table 1) we have a limited number of such baselines allof which are aligned with the N-S direction, allowing usto label the baselines with just numbers Un instead ofvectors U n. Earlier studies (e.g. Bharadwaj & Sethi 2001;Ali & Bharadwaj 2014) show that the H i signal contribu-tion to S2(Un, νi, νj) can be modelled as S2(Un,∆νij) whichdepends only on the frequency separation ∆νij = |νi−νj |. Itis further predicted that S2(Un,∆νij) has a maximum valueat ∆νij = 0, and decorrelates rapidly as the separation ∆νijincreases, with a value close to zero for large values of ∆νij(see e.g. Chatterjee et al. 2017, for result from simulation).S2(Un,∆ν) is predicted to be close to zero for ∆νij > 1MHzon nearly all of the baselines that will be probed by P-Iof OWFA (Ali & Bharadwaj 2014). This essentially impliesthat the H i signal is mainly localized near the diagonal ele-ments of the visibility correlation matrix S2(U n, νi, νj), theelements which are located far away from the diagonal areexpected to not have much of the H i signal. It is thereforereasonably safe to assume that the matrix elements locatedat a distance from the diagonal are entirely dominated bythe foregrounds with a negligible H i contribution.

In the current work we present preliminary resultswhere we explore the frequency dependence of S2(U n, νi, νj)in order to assess the contribution to such effects from theinstrument itself. We note that an earlier version of the vis-ibility correlation-based estimator presented here has beenapplied to measure the foreground Cℓ in 150MHz GMRTobservations (Ali et al. 2008; Ghosh et al. 2012), and alsofor foreground removal in 610MHz GMRT observations(Ghosh et al. 2011a,b).

5.2 Computing the estimator

In practice, we would calibrate the observed visibilities mak-ing use of the redundant spacings (e.g. Wieringa 1992;Liu et al. 2010; Marthi & Chengalur 2014). In this simula-tion we directly use the model visibilities, equivalent to theobserved visibilities having been already calibrated. For abaseline U n and a frequency channel νi, let there be Nn

distinct measurements of the visibility V(a)(U n, νi) labelledusing a = 1, 2, ..., Nn. Note that each redundant spacing aswell as each time sample is a distinct measurement of thevisibility3. The Nn distinct measurements all measure thesame Fourier mode on the sky, given by the model visibili-ties M(U n, νi). The system noise contribution N a (U n, ν)

3 This is true only when the telescope tracks the sky continuously.We do not consider the drift-scan mode of observing in this article.

0.01

0.1

1

10

100

1 10 100

Jy

U (λ)

0.01

0.1

1

10

100

1 10 100

Jy

U (λ)

0.01

0.1

1

10

100

1 10 100

Jy

U (λ)

0.01

0.1

1

10

100

1 10 100

Jy

U (λ)

100 200 300 400 500

U (λ)

100 200 300 400 500

U (λ)

100 200 300 400 500

U (λ)

100 200 300 400 500

U (λ)

Figure 7. (colour online) The red, blue, black and magentacolours code respectively the declinations 0, 20, 40 and 60.This figure shows the magnitude of the model visibilities for a sin-gle realization of the sky, the emission from the larger scales beingpicked up by the shortened baselines at increasing declination.

(equation 11), however, is different in each of these measure-ments: its contribution to each visibility measurement is anindependent Gauusian random variable, and the differentvisibility measurements can be added coherently (after cali-bration) to improve the signal-to-noise ratio; this is a uniquefeature of OWFA. Therefore, the sum

V(U n, νi) =1

Nn

Nn∑

a=1

V(a)(U n, νi) (27)

shows a baseline-dependent 1/√Nn noise behaviour.

We can, in principle, use V(U n, νi)V∗(U n, νj) to esti-mate the visibility correlation S2(U n, νi, νj). The correla-tion S2(U n, νi, νi), however, will pick up an extra, positive,noise bias

∑Nn

a=1 |N a (U n, νi) |2/(Nn)2 arising from the cor-

relation of a visibility with itself. It is possible to avoid thisnoise bias (e.g. Begum et al. 2006; Choudhuri et al. 2016b)by subtracting out the self correlation

V2(U n, νi) =1

Nn

Nn∑

a=1

|V(a)(U n, νi)|2 . (28)

We hence define our estimator S2 as

S2(U n, νi, νj) =N2

n V(U n, νi)V∗(U n, νj)− δijNn V2(U n, νi)

N2n − δij Nn

(29)

which is free from the noise bias arising from the correlationof a measured visibility with itself.

6 RESULTS

6.1 Visibilities

The model visibilities are computed for a sky model thatconsists of the sum of a realization of the diffuse foregroundand the extragalactic radio sources. The full U -range of theOWFA is ∼ 0− 500λ at δ0 = 0. However, for fields at highdeclinations, the range of U is compressed by the factorcos δ0, where δ0 is the declination of the centre of the field.The model visibilities for a single realization of the sky that

MNRAS 000, 1–16 (2017)

Foreground predictions for OWFA 11

includes both the diffuse emission and extragalactic radiosources for four different declinations are shown in Figure 7.The higher declinations allow for a compressed range of U ,thereby accessing more of the extended emission at U <100. However, the sensitivity reduces equally by cos δ0 dueto the smaller projected aperture.

6.2 The Foreground Power Spectrum

In this section we provide results for the visibility correla-tion MAPS estimator. As a verification step, we start with anoise-free simulation, and use the model visibilities directly:the assumption of perfect calibration and absence of iono-sphereic effects and RFI is implicit. The MAPS estimatorreturns a matrix of the visibility correlation S2(U n, νi, νj)as a function of the pair of frequencies at which it is com-puted. Each panel (left to right) in Figure 8 is the matrix(read top left to bottom right) S2(U n, νi, νj) for the OWFAbaselines n = 1, 16 and 32 respectively.

This observation is simulated for a declination of δ = 0

and includes the diffuse Galactic emission as well as thepoint source contribution. The three panels of Figure 8 (leftto right), have different (decreasing) foreground amplitudes.As discussed in Figure 9 of Ali & Bharadwaj (2014), the dif-fuse Galactic emission is the dominant component at short-est baseline as apparent in the left panel of the Figure 8. Theclustered point source component dominates at small angu-lar scales or the larger baselines (U . 300) which can beseen in the middle panel. The foreground amplitude rightpanel where U ≃ 400, is dominated by the Poisson con-tribution. The foregrounds appear to be quite smooth inthe νi − νj plane, suggesting that this could possibly beused to separate the H i signal, which has a different signa-ture in S2, from the foregrounds. Note that the foregroundsexhibit a particularly smooth frequency dependence at thesmallest baseline (left panel), and the frequency variationsbecome progressively more rapid as we move to larger base-lines. Much of the frequency structure seen in the centraland right panels is clearly not a function of the frequencyseparation ∆ν = |νi − νj | alone, and these structures arevisible even at large ∆ν, or away from the diagonal. Thisdistinction between the foregrounds and the H i signal leadsto separable features between the foregrounds and H i in thek-space.

We next use the diagonal elements S2(U n, νi, νi) of themeasured MAPS to estimate the angular power spectrumCℓn(νi, νj) of the sky signal using equation (25). Figure 9(a)shows Cℓ(ν) recovered from two simulated realisations of thediffuse Galactic foregrounds. The solid line is the input ana-lytical angular power spectrum and the points represent therecovered angular power spectrum for a single realisation.Similarly, Figure 9(b) shows the recovered angular powerspectrum for the extragalactic point sources, where the solidline is the input angular power spectrum, which is the sum ofthe Poisson and the clustered contributions. We see that therecovered angular power spectrum is in excellent agreementwith the input model for the entire ℓ range shown in thefigure. In both cases, the dashed line represents the modelused to obtain a random realization, and serve only as avisual aid. We reiterate that the results are shown here fora noiseless simulation. The MAPS estimator, in addition,is unbiased as it avoids the self-noise contribution through

equation (29). In a future article, we will explore apply-ing the tapered gridded estimator (TGE; Choudhuri et al.2014), tailoring it suitably for OWFA.

7 DISCUSSION

We will now discuss some properties of the estimator inthe context of the OWFA experiment, including limitationsposed by the instrument systematics. We will look at thefollowing aspects in succession:

• The estimator shows predictable spectral behaviour andintroduces no spectral anomalies.

• The chromatic primary beam and the chromatic re-sponse function of the interferometer dominate well over thechromaticity of the measured sky signal.

• Contribution from sources in the sidelobes manifest asspectral signatures in the estimator.

• The error on the estimator can be derived from thedata.

7.1 Spectral behaviour

The MAPS estimator has a fully tractable spectral response.Since it is possible to analytically derive the spectral re-sponse of the estimator, we can ascertain through simula-tions that the estimator introduces no unaccounted featuresin the spectrum. This means that, in principle, it is possibleto model out the instrument-induced spectral features in theestimator.

It has been shown that the angular power spectrumCℓ(ν) and the MAPS estimator S2(U , νi, νj) are related byequation (24), where for our assumed sinc-squared primarybeam,

A(U, ν) =λ2

bdΛ

(

uλ

d

)

Λ

(

vλ

b

)

(30)

is the Fourier transform of the primary beam power pattern,U = (u, v) and Λ(x) is the triangular function defined as

Λ(x) = 1− |x| for |x| < 1, and Λ(x) = 0 for |x| > 1. (31)

Given that ℓ = 2πU where U = d/λ, we may write

Cℓ ∝(

λ

d

)γ

(32)

It then follows that for the visibility correlation of the diffuseforeground signal alone,

S2 ∝ ν2(1−α)−γ (33)

considering all the spectral contributions, namely the in-trinsic spectral index α, the power law index dependenceon frequency through ℓ, and the Fourier transform of theprimary beam power pattern.

Restating that the spectral response of the estimatoris fully tractable, we illustrate this for a simple case whereα = 2.32 and γ = 2.54. The S2 matrix is obtained for 1000independent but noiseless realisations of the diffuse Galacticforeground and the mean and the 1σ error bars of these di-agonals are computed. The mean and the error bars are thenindividually normalised at ν = 326.5 MHz; the normalisedmean and error bar curves can be represented as ‖S2(ν)‖

MNRAS 000, 1–16 (2017)

12 Marthi et al.

Figure 8. (colour online) The MAPS estimator S2(U n, νi, νj) for a single realization of the sky, for simulated observations at δ = 0 forOWFA baselines n = 1, 16 and 32 increasing from left to right panel. The diagonal in each matrix is the locus of ∆ν = 0, and any pointon the diagonal through the MAPS for the baselines stacked in sequence is a proxy for Cℓ(∆ν = 0). In addition to the shorter baselineshaving more power, the longer baselines exhibit enhanced spectral structure, both in line with our understanding. See text for details

.

(a) Diffuse foreground (b) Point source foreground

Figure 9. (a) The angular power spectrum of the diffuse Galactic emission: input model shown with the solid line and the powerspectrum recovered through the estimator with points, for two random realisations. (b) The angular power spectrum of extra-galacticpoint sources. In both the plots, the dashed line represents the analytical expression for the power spectra used to obtain the randomrealisations, shown here only to compare the random realisations against. The results are at the frequency νc = 326.5 MHz.

and ‖∆S2(ν)‖. Figure 10 shows the normalised mean of thediagonals as filled black circles. These have been sampledat every tenth channel so that they are seen clearly in theplot. The shaded region represents the bounds of the 1σ er-ror on the mean of the foreground realisations. The curvesin red are the normalised (at 326.5 MHz) S2(ν) curves fortwo different realisations. The black curve is the mean ofall the realisations and it agrees remarkably well with the

expression(

νν0

)2(1−α)−γ

. We make a few points to enable a

clear interpretation of Figure 10. Firstly, the error boundson the analytical curve arise from the unit variance complexrandom variable (x + iy)/

√2 in equation (14). Therefore,

it shows the standard deviation of the random realisationsand not the estimator. The standard deviation computedfrom the realisations is indeed unity, which is easy to see atνc = 326.5 MHz. In addition, the red curves show just twoof such random realisations, and there is visibly more spec-tral structure on the longer baselines. As a result of which,

MNRAS 000, 1–16 (2017)

Foreground predictions for OWFA 13

Figure 10. (colour online) The normalised MAPS estimator ‖S2(U , ν)‖ is shown left to right, top to bottom, for baselines 4, 21 and31. The red curves relate to two different realisations of the diffuse Galactic synchrotron foreground from a total of 1000. The estimatedmean is given by the filled circles and the solid black line is the analytically computed curve given by (ν/ν0)(2−2α−γ) . The simulationis over 39 MHz split into 312 channels, but only a sampled version of the estimated mean (filled circles) is shown for clarity. The shadedregion represents the 1σ error bound on the random foreground realisations.

21.2

21.3

21.4

21.5

21.6

21.7

21.8

0 5 10 15 20 25 30 35 40

S2

Jy2

∆ν MHz

2.5

3

3.5

4

4.5

5

5.5

6

0 5 10 15 20 25 30 35 40

S2

x 1

0-2 J

y2

∆ν MHz

-2

-1

0

1

2

3

4

5

0 5 10 15 20 25 30 35 40

x 1

0-6 J

y2

∆ν MHz

-4

-3

-2

-1

0

1

2

3

0 5 10 15 20 25 30 35 40

x 1

0-9 J

y2

∆ν MHz

Figure 11. The S2(∆ν) curves for two baselines U 2 and U 17 areshown in the top left and right panels respectively. The residualafter fitting polynomials are shown in the bottom panels, indicat-ing the contribution from sources in the sidelobes.

lastly, the match is not visibly exact for the longer baselinebetween the analytical curve and the estimated mean datapoints. A progressively larger number of realisations wouldhave to be averaged for the match to appear exact on thelonger baselines. In addition, the shaded bounds allow for acomparison of these two realisations with the error on themean, clearly showing more variance on the longer baselines.

A source away from the phase centre leaves oscillatoryfeatures in the MAPS estimator, and the frequency of theoscillation depends on the location of the source m as wellon the baseline nd. It is interesting to note that ∆ν/ν0 isof the order of 12% (∼40 MHz/327 MHz) for OWFA. Theargument can be generalised to include many point sources,and by extension to the diffuse foregrounds as well. Thesum total contribution from all emission within the fieldof view superimpose with a range of phases that tend to

partly cancel out. But residual features remain imprintedon the innocuous-looking smooth spectra. Figure 11 showsthe S2 for the same realisation of diffuse and point sourceforegrounds used to obtain the plots in Figure 8, but com-puted as a function of ∆ν by averaging along the diagonalsof the S2 matrix for two example baselines. Although theforegrounds are non-stationary as discussed in Section 5.2,it is instructive to cast the estimator in the familiar form asa function of ∆ν for this exercise. The apparently smoothcurves in the top panels can be fitted by low-order poly-nomials successively. The residual contamination can stillbe seen in the bottom panels. The frequency of the oscil-latory pattern is higher as expected for longer baselines. Itmust be noted that the amplitude of the residual oscillatoryfeatures is about 7 orders smaller than the visibility corre-lation S2. These features are still about 1-2 orders abovethe amplitude of the expected H i visibility correlation (seeAli & Bharadwaj 2014; Chatterjee et al. 2017). At this level,these dominant residual features preclude detecting the H i

signal in the ∆ν space.These undesirable oscillatory features can be suppressed

by restricting the total FoV. This can be achieved by taper-ing the primary beam with a weighting function, typicallya Gaussian window function. However, this is possible onlyin the uv-plane post-correlation, through a 2D convolution.Beam tapering in uv has been shown to be quite effective forthe GMRT (Ghosh et al. 2011b). The tapered-gridded esti-mator has been shown to be equally effective for simulatedGMRT data (Choudhuri et al. 2014, 2016a).

The function S2(U, ν1 − ν2) behaves smoothly awayfrom and normal to the principal diagonal. Since this cutnormal to the diagonal represents S2(U,∆ν), the MAPS es-timator matrix S2 naturally encodes the decorrelation of theforeground as well as the sky signal.

7.2 Sources in the sidelobes and instrument

chromaticity

The effects of sources at large angular distances from thepointing centre coupling in through the primary beam is

MNRAS 000, 1–16 (2017)

14 Marthi et al.

Figure 12. (colour online) The estimator S2(ν) for different values of α and γ are shown here for the longest baseline |U | = 475.0.Each plot has three curves; one each for α = 0.0, 1.0, 2.0, shown by the colours blue, red and purple respectively. The values have beennormalised to S2(ν) at ν = 326.5 MHz. The normlized curves look very similar between widely different values for α and γ, implying thatthese estimates are dominated not by α and γ, but by the instrument response. The dashed lines mark νc = 326.5 MHz and ||S2(νc)|| = 1.

well-known (see e.g. Datta et al. 2010; Vedantham et al.2012; Pober et al. 2013b; Thyagarajan et al. 2015a,b;Pober et al. 2016). Consider a single point source at the co-ordinates θ = (l,m). Let us also assume that the pointingof observation is (α0 = 0, δ0 = 0) without loss of general-ity. For the nth baseline in the linear array, we note that atfrequency ν = c/λ,

U n = nU 1 = nd

λ= n

d

λ0

λ0

λ(34)

The visibility, obtained as a Fourier sum in the simulationand given in equation 10, simplifies to

M (U , ν) =∑

θ

I(θ, ν) A(θ, ν) e−i2πmnd

λ0

λ0

λ (35)

Since V(U , ν) represents an estimate of M (U ), the two-visibility correlation at (νi, νj) for this baseline becomes

S2(U n, νi, νj) ∼ |I(θ, ν0)|2 |A(θ, ν0)|2 e−i2πm

(

ndλ0

)

(

νi−νjν0

)

(36)

assuming that the approximations

|I(θ, ν0)|2 = I(θ, νi)I∗(θ, νj) (37)

and

|A(θ, ν0)|2 = A(θ, νi)A∗(θ, νj) (38)

are reasonable, but in general not strictly true.The intrinsic frequency dependence of the sky signal

can be disentangled from the chromatic instrument responseby varying α and γ over a range of values. We observe theresulting spectral behaviour S2 for a single Gaussian realiza-tion of the diffuse foreground. Figure 12 shows S2(∆ν = 0)normalised to its value at ν = 326.5 MHz, for the longestbaseline, where α is allowed to take three different values:0.0, 1.0 and 2.0 (the three curves in each panel) for eachγ = 0, 1, 2. The spatial index γ exerts a spectral dependencethrough ν−γ , and the spectral index α through ν2(α−2).Therefore for γ = 0 and α = 2 (spectral index in tem-perature units), all the spectral behaviour that manifests

in the curve is indeed being caused by (a) the baselines U

changing with frequency and (b) the primary beam shrink-ing with increasing frequency. The strikingly similar curvesfor different values of α and γ point to their effect beingsub-dominant. Therefore, the bulk of the spectral trend ofS2 in a single foreground realisation comes from the chro-matic response of the instrument(see also Vedantham et al.2012; Thyagarajan et al. 2013).

7.3 Error on the estimator

Having studied the visibility correlation estimator throughnoise-free simulations, we could now turn our attention tothe effects of noise. The estimator S2 has the dimensions ofvariance. For the error on S2 we are hence interested in thevariance of the variance. In the simulations described above,the visibility correlation matrix S2 is computed at every one-second interval. For an N-second observation, therefore, Nsuch matrices are available for each baseline, from which themean matrix and the RMS matrix for each baseline can becomputed.

The estimator S2 and the error matrix ∆S2 are com-puted now for a single realisation of the Galactic dif-fuse foreground, with the system temperature of 150 K(Selvanayagam et al. 1993) and a 60-second observation.This is equivalent to deriving the error bars from as manyindependent realisations (60) of the noise, given a singlerealisation of the foreground. Figure 13 shows the recov-ered power spectrum by sampling through the mean andRMS cubes, S2 and ∆S2, at the same (νi, νj) co-ordinates:the power spectrum is obtained from the mean S2 matrixand the error bars from the RMS ∆S2 matrix. Figure 13(a)shows the “self” power spectrum with the error bars whereνi = νj =326.44 MHz, and Figure 13(b) shows the “cross”power spectrum with error bars, at νi =342.19 MHz andνj =326.94 MHz. The dashed straight line through the plotsis not a fit, but it is the input analytical power spectrumused for simulating the diffuse Galactic foreground. The U

values on the x-axis are computed at the central frequency326.5 MHz, as there is no other meaningful way to represent

MNRAS 000, 1–16 (2017)

Foreground predictions for OWFA 15

(a) Self-power spectrum (b) Cross-power spectrum

Figure 13. The panels (a) and (b) respectively show the “self-power spectrum” and “cross-power spectrum” at different values of(νi, νj) by sampling through the S2 cube. The error bars, drawn from the same co-ordinates of the RMS cube, correspond to 60 secondsof integration with a system temperature of Tsys = 150 K. The dashed straight line is the input angular power spectrum of the Galacticdiffuse synchrotron foreground. The scatter in the estimated power spectrum is due to the stochastic nature of a single realization of theforeground.

U at a pair of frequencies (νi, νj) at which these spectrahave been extracted. Figure 13 is an important result purelyfrom the perspective of foregrounds: a 60-second integrationhas resulted in a 10σ detection of the diffuse Galactic fore-grounds, under the implicit assumption that the point sourceforegrounds have been modelled out, or are sub-dominant aswould be the case for the short baselines of OWFA. This is avery optimistic scenario if we are interested in the statisticsof the diffuse foregrounds.

8 SUMMARY

In this paper we have briefly described the upcoming OotyWide Field Array interferometer, one of whose goals is todetect the large scale structure in redshifted 21-cm emissionfrom z ∼ 3.35. A detailed simulation of the instrument andthe foregrounds has been carried out. The simulations incor-porate a fully chromatic instrument model and hence leadto conservative estimates for the expected systematics. Wehave recast the visibility correlation estimator, previouslyapplied to GMRT observations, for OWFA. We find thatthe foregrounds manifest with very smooth spectra in theestimator, and that fitting them out with polynomials stillleaves systematic residuals that are 1-2 orders stronger thanthe cosmological signal. Besides, the chromatic response ofthe instrument gives rise to dominant spectral features in thebandpass of the baselines. The short baselines make OWFAa sensitive instrument for rapidly characterising the diffuseforegrounds. In a future paper, we will address further sup-pression of the foreground residuals.

9 ACKNOWLEDGEMENTS

VRM thanks CTS, IIT Kharagpur, and SC thanks NCRAfor hosting them during a part of this work. SC is supportedby a University Grants Commission Research Fellowship.The Ooty Radio Telescope is located at Radio AstronomyCentre, Udhagamandalam and operated by the NationalCentre for Radio Astrophysics, Tata Institute of Fundamen-tal Research.

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