A STUDY OF FUNDAMENTAL LIMITATIONS TO STATISTICAL DETECTION OF REDSHIFTED H i FROM THE EPOCH OF REIONIZATION

Precise measurements of the cosmic microwave background (CMB) anisotropies have constrained the background cosmology and initial conditions for structure formation. However, understanding the nonlinear growth of density perturbations and astrophysical evolution in the epoch of reionization (EoR) has been difficult. Evidence to date suggests a complex reionization history (Haiman & Holder 2003; Cen 2003; Sokasian et al. 2003; Madau et al. 2004); for instance, the CMB data, when fitted to models of instantaneous reionization, point to a reionization redshift of z ≈ 10.5–11 (Kogut et al. 2003; Jarosik et al. 2011; Komatsu et al. 2011; Larson et al. 2011), which is in conflict with observations of Gunn–Peterson absorption troughs and near-zone transmission toward distant quasars indicating rapid evolution in the ionization fraction as late as z ≈ 6–7 (Becker et al. 2001; Djorgovski et al. 2001; Fan et al. 2002; Mortlock et al. 2011).

Direct observation of redshifted 21 cm spin transition of neutral hydrogen has been identified to be a useful method for detecting structures in cosmological gas at high redshifts (Sunyaev & Zeldovich 1972; Scott & Rees 1990; Madau et al. 1997; Tozzi et al. 2000; Iliev et al. 2002). Tomography of the redshifted 21 cm line promises to be a key probe of reionization history (Zaldarriaga et al. 2004). Observing images of the three-dimensional distribution of neutral hydrogen temperature fluctuations in excess relative to the CMB temperature is expected to reveal the epoch as well as the process of reionization in detail; however, Furlanetto & Briggs (2004) point out that such imaging requires the sensitivity of the Square Kilometer Array (SKA). Recently, through the use of simulations, the potential of SKA precursors for direct imaging and detection of ionized regions during late stages of reionization has been demonstrated (Zaroubi et al. 2012; Malloy & Lidz 2013). Numerous first-generation radio telescopes such as the Murchison Widefield Array (MWA; Lonsdale et al. 2009; Tingay et al. 2013), the Low Frequency Array (van Haarlem et al. 2013), and the Precision Array for Probing the Epoch of Reionization (PAPER; Parsons et al. 2010) are becoming operational with enough sensitivity for a statistical detection of the EoR H i power spectrum. Measuring the H i power spectrum and its cosmological evolution is a first step to understanding structure formation and astrophysics in the EoR.

Power spectrum measurements of the redshifted 21 cm from EoR are difficult for the following reasons. The EoR signal is extremely weak relative to the foreground emission of the Galaxy and extragalactic sources (Bernardi et al. 2009; Ghosh et al. 2012). Considerable effort is required to distinguish their signatures from residual errors even after careful spectral modeling and subtraction of these foregrounds (Di Matteo et al. 2002; Zaldarriaga et al. 2004). Morales & Hewitt (2004) show that the inherent isotropy and symmetry of the EoR signal in frequency and spatial wavenumber (k) space make it distinguishable from sources of contamination which lack such symmetry. But they note that such symmetry considerations provide only an additional tool for separating foreground contamination from the signal and do not guarantee that foreground contamination will be removed.

An inherent mechanism of foreground contamination via the frequency-dependent structure (chromaticity) of the primary and synthesized beams has been pointed out by Bowman et al. (2009) and Morales et al. (2012). The chromatic nature of the primary and synthesized beams carries the transverse structure of contamination due to the residuals of continuum foreground subtraction into the line-of-sight direction. This has been termed mode-mixing. Both analytic calculations of Vedantham et al. (2012) and simulations of Datta et al. (2010) and Trott et al. (2012) have shown that foreground contamination by residuals after source subtraction is predominantly localized to a wedge-like region in k-space. The region excluded by the wedge has been termed the EoR window (Morales et al. 2012; Vedantham et al. 2012). They have also indicated that appropriate choices of bandpass window functions and imaging algorithms can significantly minimize levels of such contamination in specific regions of k-space.

In this paper, we present a unified framework for estimating three fundamental sources of uncertainty, namely, foreground contamination, thermal noise, and sample variance in k-space. We apply this general understanding to the case of MWA using different observing modes. We have also explored the effects of shaping the bandpass window and refining the EoR window. With detailed estimates, we compare the relative magnitudes of different sources of uncertainties and obtain a more complete view of the EoR sensitivity of the 128-tile MWA.

The rest of the paper is organized as follows. Section 2 provides a quick snapshot of the cosmology that motivates radio observations. Section 3 sets up the basic radio interferometer measurements of signal and uncertainties. Parameters and notations used are introduced that bridge the radio interferometer measurements and cosmological motivations. Section 4 introduces the framework upon which we build our understanding and estimates of different sources of uncertainties. Here, we also list some assumptions that have gone into our study. In Sections 5 and 6, we describe the k-space occupancy and estimates of foreground and thermal noise components, respectively, in the power spectrum. Section 7 provides estimates of the EoR H i power spectrum and sample variance. The detailed interplay between various uncertainties under different observing modes and instrument parameters in determining sensitivity of the instrument for statistical measurements of EoR signatures is discussed in Section 8. The results are then summarized in Section 9. In Appendices A and B, we provide the details behind the derivation and estimation of classical radio source confusion and power spectrum of extragalactic point-like sources in k-space, respectively.

Lidz et al. (2008) provide a basis for understanding the power spectrum of the 21 cm brightness temperature (relative to CMB) fluctuations in the limit that the spin temperature, T_S, is globally much larger than the CMB temperature, T_CMB. Ignoring peculiar velocities, the 21 cm brightness temperature relative to the CMB at spatial position $\overline{\mathbf {r}}$ is

$$\begin{equation} \delta _T(\overline{\mathbf {r}}) = T_0\,\langle x_{{\rm H}} \rangle \,[1+\delta _x(\overline{\mathbf {r}})]\,[1+\delta _\rho (\overline{\mathbf {r}})]. \end{equation} \tag{ 1 }$$

Here, T₀ is the 21 cm brightness temperature of a neutral gas element with cosmic mean gas density, at redshift z, observed at frequency f₀ = 1420/(1 + z) MHz, relative to the CMB temperature at that epoch. T₀ = 28 [(1 + z)/10]^1/2 mK for the cosmological parameters we have adopted throughout this paper (symbols have their usual meanings): H₀ = 70 km s⁻¹ Mpc⁻¹, Ω_m = 0.27, Ω_Λ = 0.73, and $\Omega _K=1-\Omega _m-\Omega _\Lambda$. 〈x_H〉 is the volume-averaged neutral fraction, δ_x is the fractional fluctuation in the neutral fraction, and δ_ρ is the fractional gas density fluctuation. The volume-averaged ionization fraction is 〈x_i〉 = 1 − 〈x_H〉.

The power spectrum of $\delta _T(\overline{\mathbf {r}})$ is given in k-space by $P^{{\rm H\,{\scriptsize I}}} (\overline{\mathbf {k}})$, which is the Fourier transform of $\left \langle \delta _T(\overline{\mathbf {r}})\,\delta _T(\overline{\mathbf {r}}+\Delta \overline{\mathbf {r}})\right \rangle$, and $\overline{\mathbf {k}}$ is the Fourier conjugate variable of $\Delta \overline{\mathbf {r}}$. Assuming isotropy of neutral hydrogen distribution, $P^{{\rm H\,{\scriptsize I}}} (\overline{\mathbf {k}})$ may be described using only the radial coordinate k, as $P^{{\rm H\,{\scriptsize I}}}(k)$. Equivalently, the dimensionless quantity $\Delta _{\rm H\,{\scriptsize I}}^2(k)$ is frequently used to represent power in a logarithmic interval of k, given by (Zaldarriaga et al. 2004; McQuinn et al. 2006; Lidz et al. 2008)

$$\begin{equation} \Delta _{\rm H\,{\scriptsize I}}^2(k)=k^3P^{{\rm H\,{\scriptsize I}}}(k)/2\pi ^2T_0^2. \end{equation} \tag{ 2 }$$

$P^{{\rm H\,{\scriptsize I}}}(k)$ is estimated using the image cube, I(l, m, f), representing the sky brightness distribution in (l, m, f)-coordinates. In radio interferometry, I(l, m, f) is obtained by Fourier transforming the visibility measurements, V(u, v, f), made in (u, v, f)-coordinates. u and v are baseline lengths in units of wavelength, and l and m denote the direction cosines on the celestial sphere. f denotes the frequency of observation. η represents instrumental delay. (l, m, f) and (u, v, η) form a Fourier conjugate pair of variables. We adopt the following convention for Fourier transform:

$$\begin{eqnarray} &&V(u,v,\eta) = \int \int \int I(l,m,f) e^{-j2\pi (ul+vm+\eta f)}\,{d}l\,{d}m\,{d}f, \quad \end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray} I(l,m,f) &= \int \int \int V(u,v,\eta)\,e^{j2\pi (ul+vm+\eta f)}\,{d}u\,{d}v\,{d}\eta, \quad \end{eqnarray} \tag{ 4 }$$

where $j=\sqrt{-1}$.

The sky brightness distribution comprising the true EoR H i signal and foregrounds is multiplied by the primary beam power pattern, $W_{lm}^{\rm P}(l,m)$, on the sky. Equivalently, the visibilities of true EoR H i signal, $V_{uvf}^{\rm H\,{\scriptsize I};T}(u,v,f)$, and foregrounds, $V_{uvf}^{\rm FG;T}(u,v,f)$, are convolved with the spatial frequency response of the power pattern of an individual antenna, $W_{uv}^{\rm P}(u,v)$. $W_{lm}^{\rm P}(l,m)$ and $W_{uv}^{\rm P}(u,v)$ form a Fourier transform pair. The convolved visibilities are corrupted by additive thermal noise, $V_{uvf}^{\rm N}(u,v,f)$, sampled at the baseline locations given by the sampling function, S_uv(u, v). The sampling function is the Fourier counterpart of the synthesized beam, S_lm(l, m). Along the line of sight, the visibilities are modified by the frequency bandpass weights, $W_f^{\rm B}(f)$. Thus, the measured visibilities may be expressed as

$$\begin{eqnarray} && V_{uvf}^{\rm obs}(u,v,f) = \left \lbrace \left[V_{uvf}^{\rm H\,{\scriptsize I};T}(u,v,f)+V_{uvf}^{\rm FG;T}(u,v,f)\right]\right. \nonumber \\ &&\left. \quad \ast W_{uv}^{\rm P}(u,v) + V_{uvf}^{\rm N}(u,v,f)\right \rbrace \,S_{uv}(u,v)\,W_f^{\rm B}(f), \end{eqnarray} \tag{ 5 }$$

where the symbol * denotes convolution. Equation (5) forms the basis of our estimates of signal and various components of uncertainty in the power spectrum. A matrix-based framework is described in Liu & Tegmark (2011).

In Fourier space, $V_{uvf}^{\rm obs}(u,v,f)$ is transformed to

$$\begin{eqnarray} V_{uv\eta }^{\rm obs}(u,v,\eta) &= \int V_{uvf}^{\rm obs}(u,v,f)\,e^{-j2\pi \eta f}{d}f. \end{eqnarray} \tag{ 6 }$$

This is obtained by Fourier transforming along frequency.

The characteristic size of the spatial frequency response of the tile's power pattern, $W_{uv}^{\rm P}(u,v)$, is A_e/λ², where A_e is the effective area of a tile and λ is the observing wavelength. The Fourier response of the bandpass window, $W_f^{\rm B}(f)$, is $W_\eta ^{\rm B}(\eta)$. The characteristic width of the bandpass response function, $W_\eta ^{\rm B}(\eta)$, is set by the inverse of effective bandwidth, B_eff. $W_{uv\eta }(u,v,\eta)=W_{uv\eta }^{\rm P}(u,v,\eta)\ast W_\eta ^{\rm B}(\eta)$ is the instrumental response in the spatial frequency domain, where $W_{uv\eta }^{\rm P}(u,v,\eta)$ may be interpreted as the spatial frequency response of the tile's power pattern obtained over a uniform and infinite bandpass window, and $W_{uv}^{\rm P}(u,v)=W_{uv\eta }^{\rm P}(u,v,\eta =0)$. Here, we have assumed that the power pattern of the tile does not vary significantly over the chosen frequency band. W_uvη(u, v, η) is set such that ∫W_uvη du dv dη = 1, where δu δv δη ≃ A_e/(λ²B_eff). In our adopted Fourier convention, the primary beam, $W_{lm}^{\rm P}(l,m)$, and bandpass window function, $W_f^{\rm B}(f)$, each have peaks of value unity.

True sky visibilities are uncorrelated between non-identical baseline vectors (spatial frequencies). W_uvη(u, v, η) is an instrumental function that introduces correlations between true sky visibilities. $W_{uv}^{\rm P}(u,v)$ and $W_\eta ^{\rm B}(\eta)$ are the transverse and line-of-sight components, respectively, of this correlating instrumental function. $W_{uv}^{\rm P}(u,v)$ is a convolution of the electric field distribution over the tile with itself, which is, alternatively, the Fourier transform of the power pattern of the tile. If the electric field pattern of the tile was a uniform square, $W_{uv}^{\rm P}(u,v)$ takes the shape of a square pyramid in the (uv)-plane.

We use the array configuration of MWA to estimate the signal and uncertainties. For such an estimation, we gridded the (uv)-plane with a cell size of λ/2 on each side. This is to avoid aliasing effects in the image up to the spatial scale of the horizon, (l, m) ∈ [ − 1, +1]. $W_{uv}^{\rm P}(u,v)$ is obtained on the grid by the convolution described above. We assume each MWA array element to be a 4 × 4 array of identical radiators. This results in a discrete form of the square pyramid for $W_{uv}^{\rm P}(u,v)$. Some authors (McQuinn et al. 2006; Bowman et al. 2006, 2007; Beardsley et al. 2013) have assumed that W_uvη(u, v, η) is a sharply peaked function, and hence approximated it by a delta function. In our study, we use the full functional form. It is important to keep this functional form in order to take into account the multi-baseline mode-mixing, which has been described in Hazelton et al. (2013). The shape of $W_{uv}^{\rm P}(u,v)$ we use is also consistent with the baseline power response shown in Hazelton et al. (2013).

We place the synthesized baselines represented by the sampling function S_uv(u, v) as a two-dimensional histogram on this grid. Compared to an optimal gridding scheme, the histogram method could place the baselines in the grid with a maximum error of λ/2. This could cause a jitter in the weights on the grid which is most severe on scales comparable to the cell size in the grid (corresponds to spatial scale of horizon or larger). While such a jitter will be important in the case of a synthesis imaging procedure, it is not as significant in determining grid weights in our study.

In describing the aforementioned radio interferometer quantities and desired cosmological measurements, we use the following notation interchangeably throughout this paper:

$$\begin{equation} \underline{\mathbf {u}} \equiv (u,v), \,\, \underline{\mathbf {l}} \equiv (l,m), \,\, \overline{\mathbf {u}} \equiv (\underline{\mathbf {u}},\eta), \,\, \mathrm{and} \,\, \overline{\mathbf {l}} \equiv (\underline{\mathbf {l}},f). \end{equation} \tag{ 7 }$$

The spatial wave vectors, $\overline{\mathbf {k}}\equiv (\underline{\mathbf {k}},k_\parallel)\equiv (k_x,k_y,k_\parallel)$, are related to $\overline{\mathbf {u}}$ as (Morales & Hewitt 2004)

$$\begin{eqnarray} \underline{\mathbf {k}} = \frac{2\pi \,\underline{\mathbf {u}}}{D(z)}, &\quad k_\parallel \approx \frac{2\pi \,\eta }{\left(\frac{c\,(1+z)^2}{H_0\,f_{21}\,E(z)}\right)}, \end{eqnarray} \tag{ 8 }$$

$$\begin{eqnarray} \mathrm{with} \quad \Delta D(z) &\approx \frac{c\,(1+z)^2}{H_0\,f_{21}\,E(z)}\,\Delta f, \end{eqnarray} \tag{ 9 }$$

where H₀ and $E(z)\equiv [\Omega _{\rm M}(1+z)^3+\Omega _k(1+z)^2+\Omega _\Lambda ]^{1/2}$ are standard terms in cosmology and D(z) is the transverse comoving distance at redshift z (Hogg 1999). f₂₁ is the rest frequency of the 21 cm line. In the last equation, ΔD(z) may be identified as the line-of-sight comoving width of the observation at redshift z if Δf is set to the observing bandwidth. The following relations may also be noted:

$$\begin{equation} k_\perp = |\:\underline{\mathbf {k}}| = \left(k_x^2+k_y^2 \right)^{1/2} \quad \mathrm{and} \; k = |\:\overline{\mathbf {k}}| = \left(k_\perp ^2+k_\parallel ^2 \right)^{1/2}. \end{equation} \tag{ 10 }$$

k_⊥ and k_∥ may be viewed as components of $\overline{\mathbf {k}}$ along the transverse and line-of-sight directions, respectively.

Figure 1 is a schematic illustration of different regions of significance in k-space. The range in k_⊥ is set by the minimum and maximum baseline lengths, while that along k_∥ is set by the channel resolution and bandwidth. This region is the instrumental window. Residuals from unsubtracted foregrounds and the structure of their frequency-dependent sidelobes occupy the wedge-shaped region labeled as "foregrounds." The part of the instrumental window excluding the wedge, called the EoR window, is also shown.

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The radio measurements described above are related to the desired cosmological quantities. The power spectrum of EoR H i fluctuations is related to the diagonal of the covariance matrix of true H i visibilities in Fourier space as (Morales & Hewitt 2004; Morales 2005)

$$\begin{eqnarray} &&P^{\rm H\,{\scriptsize I}}(\overline{\mathbf {u}}) = \left \langle V_{uv\eta }^{\rm H\,{\scriptsize I};T}(\overline{\mathbf {u}}_i)^\star V_{uv\eta }^{\rm H\,{\scriptsize I};T}(\overline{\mathbf {u}}_j)\right \rangle \; \delta _{ij} = \left \langle \left|V_{uv\eta }^{\rm H\,{\scriptsize I};T}(\overline{\mathbf {u}})\right|^2\right \rangle, \quad \end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray} \mathrm{where} \qquad P^{\rm H\,{\scriptsize I}}(\overline{\mathbf {u}}) &= P^{\rm H\,{\scriptsize I}}(\overline{\mathbf {k}})\,\left(\frac{1}{D}\right)^2\left(\frac{B_{\rm eff}}{\Delta D}\right). \end{eqnarray} \tag{ 12 }$$

The last equation is obtained using the Fourier conventions and Jacobian in the transformation between quantities in (k_x, k_y, k_∥)- and (u, v, η)-coordinates.

Following the notations established in Section 3, we start with visibility measurements, V(u, v, f), which when Fourier transformed along (u, v)-coordinates yield an image cube I(l, m, f). From I(l, m, f) we assume that the extragalactic point-like foreground sources have been perfectly removed down to a certain source confusion threshold, to obtain a residual image cube ΔI(l, m, f), which consists of unsubtracted sources and their sidelobes. A statistical representation of ΔI(l, m, f), together with the sampling in the (u, v)-plane given by the array configuration, forms the basis of our understanding of different uncertainties.

4.1. Instrument Properties and Adopted EoR Model

4.2. Case Studies of Model Observations

4.3. Assumptions

One of the major contaminants in the EoR H i signal is the synchrotron emission from extragalactic and Galactic foregrounds (Di Matteo et al. 2002; Zaldarriaga et al. 2004). Thus far, neither of these causes of foreground contamination have been accounted for in the estimates of sensitivity of three-dimensional EoR H i power spectrum in a comprehensive manner. For instance, Beardsley et al. (2013) estimated sensitivity by excluding a wedge-shaped region, thereby removing a majority of foreground contamination. But they did not consider possible spillover from this contamination. Does it imply with certainty that foregrounds play no role any more in contaminating the EoR window or in estimates of sensitivity? Detailed estimates of extragalactic foreground contamination as done below in this paper show that contamination is also present in the EoR window outside the wedge-shaped region, even after a perfect subtraction of foregrounds.

5.1. Classical Radio Source Confusion

5.2. Foregrounds in k-space

The foreground component of measured visibilities may be written from Equation (5) as

$$\begin{eqnarray} &&V_{uvf}^{\rm FG}(\underline{\mathbf {u}},f) = V_{uvf}^{\rm FG;T}(\underline{\mathbf {u}},f) \ast W_{uv}^{\rm P}(\underline{\mathbf {u}})\,S_{uv}(\underline{\mathbf {u}})\,W_f^{\rm B}(f). \end{eqnarray} \tag{ 13 }$$

Thus,

$$\begin{eqnarray} V_{uvf}^{\rm FG}(\overline{\mathbf {u}}) &= V_{uv\eta }^{\rm FG;T}(\overline{\mathbf {u}})\,S_{uv}(\underline{\mathbf {u}}) \ast W_{uv}^{\rm P}(\underline{\mathbf {u}}) \ast W_\eta ^{\rm B}(\eta), \end{eqnarray} \tag{ 14 }$$

where the true foreground visibilities, $V_{uvf}^{\rm FG;T}(\underline{\mathbf {u}},f)$, have been convolved with the spatial frequency response of the tile's power pattern, $W_{uv}^{\rm P}(\underline{\mathbf {u}})$, multiplied by the sampling function in the (u, v)-plane, $S_{uv}(\underline{\mathbf {u}})$, multiplied by the frequency bandpass window function, $W_f^{\rm B}(f)$, and Fourier transformed along f.

The power spectrum of foregrounds is extracted from the diagonal of the covariance matrix $\langle V_{uv\eta }^{\rm FG}(\overline{\mathbf {u}}_i)^\star V_{uv\eta }^{\rm FG}(\overline{\mathbf {u}}_j) \rangle$. After certain algebraic simplifications between quantities in Fourier ($\overline{\mathbf {u}}$)- and real ($\overline{\mathbf {l}}$)-space, the foreground component of the power spectrum may be expressed as

$$\begin{eqnarray} P^{\rm FG}_{\rm inst}(\overline{\mathbf {u}}) &=& \int \int P_{lm}^{\rm FG}(\underline{\mathbf {l}})\,\ast \,|S_{lm}(\underline{\mathbf {l}})|^2\, |W_{lm}^{\rm P}(\underline{\mathbf {l}})|^2\nonumber \\ &&\!\!\ast \left|W_\eta ^{\rm B}(\eta)\right|^2\,\ast \,\delta \left(\eta +\frac{\underline{\mathbf {x}}\cdot \underline{\mathbf {l}}}{c}\right)\,{d}^2 \underline{\mathbf {l}}, \end{eqnarray} \tag{ 15 }$$

where $P_{lm}^{\rm FG}(\underline{\mathbf {l}})\equiv \sigma _{\rm C}^2(\underline{\mathbf {l}})$. The classical source confusion variance estimated across the image is weighted by primary beam squared $|W_{lm}^{\rm P}(\underline{\mathbf {l}})|^2$, then convolved with the synthesized beam squared $|S_{lm}(\underline{\mathbf {l}})|^2$, and is finally convolved with the instrumental delay response squared $|W_\eta ^{\rm B}(\eta)|^2$, which is shifted in η by $\underline{\mathbf {x}}\cdot \underline{\mathbf {l}}/c$ as prescribed by the delta function $\delta (\eta +({\underline{\mathbf {x}}\cdot \underline{\mathbf {l}}}/{c}))$. $\underline{\mathbf {x}}\equiv \underline{\mathbf {u}}\lambda$ is the baseline vector in units of distance. The details of the derivation of Equation (15) have been laid out in Appendix B.

Equation (15) enables evaluation of foreground contamination in k-space, which depends on classical source confusion $P_{lm}^{\rm FG}(\underline{\mathbf {l}})$, the primary beam pattern of the tile $W_{lm}^{\rm P}(\underline{\mathbf {l}})$, Fourier response of bandpass weights $W_\eta ^{\rm B}(\eta),$ and synthesized beam $S_{lm}(\underline{\mathbf {l}})$. It expresses the foreground contamination that is in (u, v, η)-coordinates (Fourier space) in terms of quantities in (l, m)-coordinates (real space). This expression aids in understanding the mode-mixing aspect of foreground contamination. The delta function indicates that unsubtracted sources at different locations and their sidelobes contribute predominantly to specific values of η in delay space (and corresponding k_∥-modes) given by $\eta \simeq \underline{\mathbf {x}}\cdot \underline{\mathbf {l}}/c = \underline{\mathbf {u}}\cdot \underline{\mathbf {l}}/f$. This is consistent with the findings of Vedantham et al. (2012). The equation naturally leads to a wedge-shaped region, whose boundary is set by the horizon limit (Parsons et al. 2012) in (l, m)-space given by $|\:\underline{\mathbf {l}}\:|^2=l^2+m^2=1$. The equation in k-space for the horizon limit (wedge boundary) is

$$\begin{equation} k_\parallel = \frac{H_0\,E(z)\,D_{\rm M}(z)}{c\,(1+z)}\,k_\perp. \end{equation} \tag{ 16 }$$

Using a natural weighted synthesized beam, $S_{lm}(\underline{\mathbf {l}})$, primary beam, $W_{lm}^{\rm P}(\underline{\mathbf {l}})$, obtained by a phased 4 × 4 array of identical radiators, instrumental delay response $W_\eta ^{\rm B}(\eta)$ for different bandpass shapes, and source confusion variance, $P_{lm}^{\rm FG}(\underline{\mathbf {l}})$, we have estimated foreground contamination of power spectrum in full three-dimensional k-space by evaluating the integral in Equation (15). The size of a cell in three-dimensional k-space, termed voxel, corresponds to ∼A_e and ∼$B_{\rm eff}^{-1}$ in the transverse and line-of-sight directions, respectively. The k-space volume of a voxel is ≃ 9.1 × 10⁻⁸ Mpc⁻³.

In order to illustrate the three-dimensional foreground contamination in the two-dimensional (k_⊥, k_∥)-plane, we azimuthally averaged the foreground contamination using an inverse quadrature weighting as though foregrounds were the only source of uncertainty. Figure 2 shows the averaged foreground contamination (in units of K²) in k-space for a 6 hr synthesis, where the effects of band shape, $W_\eta ^{\rm B}(\eta)$, are not applied. The structure of the foreground power spectrum is found to be in agreement with the foreground window illustrated in Figure 1. The lower and upper bounds on the k_⊥-axis are provided by the minimum (≃ 3λ) and maximum (≃ 1636λ) baseline lengths, and those on the k_∥-axis are provided by ∼1/B_eff (gray horizontal line, η_min ≃ 0.125 μs) and ∼1/(2 Δf) (η_max ≃ 12.5 μs), respectively. The foreground component of the power spectrum occupies a wedge-shaped region in k-space. The horizon limit (gray line with positive slope) sets the boundary of the wedge. For the parameters listed in Table 1, the instrumental window is given by 0.0014 Mpc⁻¹ ⩽ k_⊥ ⩽ 1.47 Mpc⁻¹ and 0.048 Mpc⁻¹ ⩽ k_∥ ⩽ 4.7 Mpc⁻¹, and the slope of the foreground wedge is 3.18. The maximum foreground contamination is ≈1.5 × 10⁻⁹ K² but occurs below the instrumental window. The typical contamination close to (k_⊥, k_∥) ≃ (0.015, 0.047) Mpc⁻¹ is ∼10⁻¹⁰ K². The model EoR H i signal strength in this region is ∼3 × 10⁻¹⁰ K². An apparent increase in foreground contamination closer and parallel to the wedge is noted. This is attributed to an increase in $\sigma _{\rm C}^2(\underline{\mathbf {l}})$, which in turn is due to an increase in solid angles as $|\:\underline{\mathbf {l}}\:|$ approaches the horizon limit.

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If independent power spectra are averaged from N_fields patches of sky, the foreground contamination in the averaged power spectrum goes as ${\sim }1/\sqrt{N_{\rm fields}}$.

5.3. Role of Bandpass Shapes in Foreground Contamination

An important consequence of convolution in Equation (15) by the term $W^{\rm B}_\eta (\eta)$, the response of the bandpass shape in instrumental delay space, is to spill the contamination from foreground emission, which is restricted to the wedge, into the regions beyond. This spillover, therefore, fills even the desirable portions of k-space, namely, the EoR window. Hence, a simple removal of the wedge-shaped region from data analysis does not completely remove all the effects of foreground contamination. The level of this spillover may be controlled by appropriate choice of bandpass shapes. Vedantham et al. (2012) have discussed the possibility of using a Blackman–Nuttall window function (Nuttall 1981).

In the context of bandpass window shaping, this paper addresses the following questions: what is the effect of bandpass shaping on foreground contamination in three-dimensional k-space? And, what is its significance to overall sensitivity when other uncertainties are also taken into account?

An infinite bandpass will convolve the power spectrum of foregrounds by a delta function, resulting in zero spillover, but is impossible to achieve in practice. A more practical band shape, such as a rectangular window, will manifest as a sinc-shaped response along η (and k_∥). A Blackman–Nuttall window is known to have a much reduced sidelobe response (3–4 orders of magnitude) along η, but its peak (area under band shape) in η-space is ≈2.76 times less than that of a sinc function, which implies a loss of sensitivity. The resolution in η is also relatively poorer. However, if the effective bandwidth, B_eff, is made equal to that of a rectangular band shape by extending the standard Blackman–Nuttall window, a significant reduction in sidelobes in the response function along η may be achieved without compromising either sensitivity or resolution, compared to those from a rectangular window. We define the effective bandwidth as

$$\begin{equation} B_{\rm eff} = \int _{-B_0/2}^{B_0/2}W_f^{\rm B}(f)\,{d}f = W_\eta ^{\rm B}(0), \end{equation} \tag{ 17 }$$

where the second equality comes from the Fourier transform convention and the limits of the integral form the edges of the band.

Figure 3 shows the rectangular (dotted), standard (solid gray), and extended (solid black) versions of Blackman–Nuttall band shapes. Their effective bandwidths are 8 MHz, 2.9 MHz, and 8 MHz, respectively. Figure 4 shows, using the respective line styles, the amplitude of the respective responses, $|W_\eta ^{\rm B}(\eta)|$, of the aforementioned band shapes along η. As expected, the standard Blackman–Nuttall window has reduced sensitivity visible by its peak and is of a poorer resolution. The extended version, however, is identical in sensitivity and resolution to that of a rectangular window.

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Using $W_\eta ^{\rm B}(\eta)$ for rectangular and extended Blackman–Nuttall windows described above, the power spectrum of unsubtracted foreground sources was estimated in three-dimensional k-space using the integral in Equation (15).

Once again for purposes of illustration we averaged these power spectra in independent cells azimuthally in k-space using inverse quadrature weighting. Figures 5 and 6 show the azimuthally averaged power spectra of foregrounds (in units of K² Hz²) for rectangular and extended Blackman–Nuttall band shapes, respectively, each with an effective bandwidth of 8 MHz. The features already noted in Figure 2 are also noted here. But the bandpass effects were not applied in Figure 2. Due to the presence of the term $W_\eta ^{\rm B}(\eta)$ in Equation (15), the foreground contamination is seen to spill over into the EoR window in both cases. The spillover into the EoR window up to k_∥ < 0.1 Mpc⁻¹ is similar in both cases and is ∼10⁴ K² Hz². A Blackman–Nuttall window is expected to be superior by 7–8 orders of magnitude in reducing the spillover beyond the wedge-shaped region relative to a rectangular window because of the term $|W_\eta ^{\rm B}(\eta)|^2$. Higher levels of foreground contamination are seen due to a rectangular bandpass window in the region k_⊥ ≲ 0.03 Mpc⁻¹, k_∥ ≳ 0.1 Mpc⁻¹ when compared to that due to a Blackman–Nuttall window. In fact, this is confirmed from Figure 7, which compares the foreground power along slices at k_⊥ ≃ 0.01 Mpc⁻¹ shown in Figures 5 and 6 as gray dashed lines. In the range 0.2 Mpc⁻¹ ≲ k_∥ ≲ 5 Mpc⁻¹, the extended Blackman–Nuttall window produces a foreground contamination spillover ∼10⁻⁶ to 10⁻⁵ K² Hz², which is about 7–8 orders of magnitude smaller than that from a rectangular window.

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Are there undesirable effects of using an extended Blackman–Nuttall band shape? A wideband observation might be analyzed with a sliding window to examine for any change in EoR detection with redshift. Thus, bandwidth is not discarded when an extended Blackman–Nuttall window is deployed. But such a window uses larger total bandwidth (more channels) than a nominal rectangular window to achieve the same effective bandwidth. If there is significant cosmic evolution of the EoR signal within the band, the assumption of statistical stationarity of EoR signal could break down and lead to a dilution of measured signal power.

The thermal noise component in a sampled visibility measurement in Fourier space, from Equations (5) and (6), is

$$\begin{eqnarray} V_{uv\eta }^{\rm N}(\overline{\mathbf {u}}) &= \int V_{uvf}^{\rm N}(\underline{\mathbf {u}},f)\,S_{uv}(\underline{\mathbf {u}})\,W_f^{\rm B}(f)\,e^{-j2\pi \eta f}{d}f. \end{eqnarray} \tag{ 18 }$$

The rms of thermal noise in a measured visibility sample in a single frequency channel is given by (Morales 2005; McQuinn et al. 2006)

$$\begin{eqnarray} \Delta V_{uvf}^{\rm N}(\underline{\mathbf {u}},f) &= \frac{\lambda ^2\,T_{\rm sys}}{A_{\rm e}\sqrt{\Delta f\,t_{\rm int}}}, \end{eqnarray} \tag{ 19 }$$

where t_int is the integration time used to obtain visibility samples.

In a natural weighting scheme, the weight of a certain (u, v)-cell is proportional to the number of baselines (or measurements) in that cell. When data measured by baselines inside (uv)-cells are averaged, the thermal noise in the averaged cell visibility is inversely proportional to the square root of number of baselines sampling that cell. Thus, the power spectrum uncertainty due to thermal noise in a spatial frequency mode ($\overline{\mathbf {u}}$) may be written as (Morales 2005; McQuinn et al. 2006)

$$\begin{eqnarray} C^{\rm N}(\overline{\mathbf {u}}) &= \left(\frac{\lambda ^2\,T_{\rm sys}}{A_{\rm e}}\right)^2\frac{\epsilon B_{\rm eff}}{t_{\underline{\mathbf {u}}}}, \end{eqnarray} \tag{ 20 }$$

where is the factor arising out of bandpass weights and $t_{\underline{\mathbf {u}}}$ is the effective time of observation for a given mode, which is effectively a product of the number of baselines sampling the mode during aperture synthesis and the integration time used in producing a visibility sample.

The thermal noise power spectrum is estimated as prescribed by Morales (2005) and McQuinn et al. (2006). arises in Equation (20) from the sum of squares of bandpass weights, $W_f^{\rm B}(f)$, while determining the power spectrum. is given by

$$\begin{eqnarray} \epsilon &= \frac{\sum _{i=1}^{N_{\rm ch}}\,\left |W_f^{\rm B}(f_i)\right |^2}{\sum _{i=1}^{N_{\rm ch}}\,W_f^{\rm B}(f_i)}, \end{eqnarray} \tag{ 21 }$$

where i indexes the frequency channels in the bandpass, N_ch is the number of channels in the bandpass, and $\sum _{i=1}^{N_{\rm ch}}\,W_f^{\rm B}(f_i)\,\Delta f = B_{\rm eff}$. The value of is 1 and 0.72, respectively, for rectangular and extended Blackman–Nuttall window band shapes. It indicates that an extended Blackman–Nuttall window also reduces the thermal noise component in the power spectrum by 28%. As a result, in order to achieve a thermal noise power equal to that from an extended Blackman–Nuttall window using a rectangular window, an observation has to be ≈40% longer.

Uncertainty in power spectrum due to thermal noise is very sensitive to observing time. In each k-mode, it is inversely proportional to the number of baselines, including redundant ones that sample that k-mode during the entire synthesis. It depends on band shape that is parameterized by . In the context of our model observations listed in Table 2, the thermal noise power spectrum in Equation (20) at any $\overline{\mathbf {k}}$-mode may be rewritten as

$$\begin{eqnarray} &&C^{\rm N}(\overline{\mathbf {k}}) = \left(\frac{\lambda ^2\,T_{\rm sys}}{A_{\rm e}}\right)^2\frac{\epsilon \, B_{\rm eff}}{N_{\rm cad}\,t_{\rm int}\,N(\underline{\mathbf {k}})\,\sqrt{N_{\rm fields}}}, \end{eqnarray} \tag{ 22 }$$

where $N(\underline{\mathbf {k}})$ is the number of baselines (redundant ones included) observing the $\underline{\mathbf {k}}$-mode during a single synthesis observation of duration t_syn and N_cad = t_obs/(t_syn N_fields).

For illustration, this is azimuthally averaged in bins of $k_\perp =(k_x^2+k_y^2)^{1/2}$ by adding the thermal noise component of the power spectrum contained in cells in this bin, in inverse quadrature, to yield a two-dimensional thermal noise power spectrum, C^N(k_⊥, k_∥). In Figures 8 and 9, we show C^N(k_⊥, k_∥) for observing modes 1 and 2, respectively, using a rectangular bandpass window. In observing mode 1, the azimuthally averaged thermal noise power spectrum attains minimum (≈640 K² Hz²) at k_⊥ ≃ 0.004 Mpc⁻¹ and maximum (≈1.4 × 10⁹ K² Hz²) at k_⊥ ≃ 1 Mpc⁻¹. The thermal noise power spectra are similar in the two observing modes except that it is higher by a factor of ≈4.47 in the latter as predicted by the scaling in Equation (22). In the first observing mode, the visibilities are added coherently for a total of 1000 hr. In the second observing mode, the visibilities on each field are added coherently only for 50 hr and power spectra are estimated. These power spectra are then averaged. This increases the thermal noise component in the power spectrum by a factor $\sqrt{20}\approx 4.47$.

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The thermal noise power spectrum exhibits cylindrical symmetry about the k_∥-axis. When an extended Blackman–Nuttall window is used, the thermal noise component in the power spectrum drops by 28% ( = 0.72) relative to that from a rectangular bandpass window.

The measured H i component of Fourier space visibilities in Equation (5) in a manner similar to Equation (14) may be expressed as

$$\begin{eqnarray} V_{uv\eta }^{\rm H\,{\scriptsize I}}(\overline{\mathbf {u}}) &= V_{uv\eta }^{\rm H\,{\scriptsize I};T}(\overline{\mathbf {u}})\,S_{uv}(\underline{\mathbf {u}}) \ast W_{uv\eta }(\overline{\mathbf {u}}). \end{eqnarray} \tag{ 23 }$$

The EoR H i power spectrum measured by the instrument, $P^{\rm H\,{\scriptsize I}}_{\rm inst}(\overline{\mathbf {k}})$, is the diagonal of the covariance matrix $\langle V_{uv\eta }^{\rm H\,{\scriptsize I}}(\overline{\mathbf {u}}_i)^\star V_{uv\eta }^{\rm H\,{\scriptsize I}}(\overline{\mathbf {u}}_j) \rangle$ and is given by (Morales & Hewitt 2004; Morales 2005; McQuinn et al. 2006; Bowman et al. 2006, 2007)

$$\begin{eqnarray} P^{\rm H\,{\scriptsize I}}_{\rm inst}(\overline{\mathbf {k}}) &= \int \int \int P^{\rm H\,{\scriptsize I}}(\overline{\mathbf {u^\prime }}) \, |W_{uv\eta }(\overline{\mathbf {u}}-\overline{\mathbf {u^\prime }})|^2\,{d}^3 \overline{\mathbf {u^\prime }} \end{eqnarray} \tag{ 24 }$$

at the sampled baseline locations, and $P^{\rm H\,{\scriptsize I}}(\overline{\mathbf {u}})$ is given by Equation (12). Although some authors (McQuinn et al. 2006; Bowman et al. 2006, 2007) have approximated $W_{uv\eta }(\overline{\mathbf {u}})$ by a delta function, it will exhibit some spillover along η depending on the bandpass shape $W_f^{\rm B}(f)$. Hence, we retain the general form of the power spectrum in Equation (24) for our work.

In Equation (24), we note the convolving effect arising out of instrumental factors, thereby introducing correlations between neighboring spatial frequencies. The observed power spectrum of the signal is a modification of the true EoR H i power spectrum by instrumental parameters of observation such as primary beam and bandpass shape.

Sample variance is equal to the power spectrum (Jungman et al. 1996; McQuinn et al. 2006). If a number of independent measurements (N_fields) of power spectrum are averaged, the sample variance goes as $P^{\rm SV}(\overline{\mathbf {k}})=P^{\rm H\,{\scriptsize I}}_{\rm inst}(\overline{\mathbf {k}})/\sqrt{N_{\rm fields}}$.

$P^{\rm H\,{\scriptsize I}}(k)$, in Equation (2), represents the variance in k-space of the spin temperature fluctuations of H i relative to the CMB. As already mentioned in Section 4.1, simulations of Lidz et al. (2008) show that the variance in the ionization field peaks at a value close to 50% ionization. We choose from the family of $P^{\rm H\,{\scriptsize I}}(k)$ curves they provide the one parameterized by (〈x_i〉, z) = (0.54, 7.32) and use it as the input model in this study. 〈x_i〉 and z are the mean volume-averaged ionization fraction and redshift, respectively. Redshifted emission from H i at z = 7.32 occurs at 170.7 MHz, which is chosen as our observing frequency. We have obtained the values of power spectrum predicted for (〈x_i〉, z) = (0.54, 7.32) from the plots of Lidz et al. (2008; A. Lidz 2011, private communication). Since we required the predicted values of the power spectrum at intermediate values of k not tabulated, we used a third-order polynomial fit to interpolate the predicted power spectrum to the required values.

Two primary causes contribute to the power spectrum of EoR H i fluctuations:

• 1.  
  the underlying matter density fluctuations, and
• 2.  
  the ionized bubbles during the reionization process.

The contribution from matter density fluctuations is anisotropic due to redshift–space distortions caused by peculiar velocity effects along the line of sight (Barkana & Loeb 2005), whereas the contribution from ionization fluctuations is isotropic. In our adopted model the ionization fraction is about 50%, which indicates significant ionization. Hence, the contribution to the EoR can be assumed to be dominated by ionized bubbles rather than due to underlying matter density fluctuations (Lidz et al. 2008). Therefore, we neglect anisotropic effects arising out of peculiar velocities in our model power spectrum.

The observed power spectrum is computed from Equation (24) using the input model. It is identical for the two observing modes listed in Table 2. The observed sample variance is higher in observing mode 1 relative to observing mode 2 by a factor $\sqrt{20}\approx 4.47$.

Figure 10 shows the observed EoR H i power spectrum in (k_⊥, k_∥)-plane corresponding to observing mode 1 while employing a rectangular band shape. Bandpass shape causes a convolution with the true power spectrum along k_∥, as in the case of the foreground power spectrum. Although not shown, as expected the extended Blackman–Nuttall window causes a far lesser spillover of the EoR H i power spectrum relative to the rectangular band shape. For our paper, we term this the "signal spillover." It is caused by the same reason (bandpass window shape) that causes foreground spillover beyond the horizon limit.

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We have estimated the EoR H i power spectrum expected to be observed and individual uncertainties in three-dimensional k-space. The total uncertainty in the power spectrum in three-dimensional k-space was obtained by summing the component uncertainties,

$$\begin{equation} \Delta P(\overline{\mathbf {k}}) = P^{\rm FG}_{\rm inst}(\overline{\mathbf {k}}) + C^{\rm N}(\overline{\mathbf {k}}) + P^{\rm SV}(\overline{\mathbf {k}}). \end{equation} \tag{ 25 }$$

8.1. Family of EoR Windows

By knowing the occupancy of various uncertainties in k-space and excluding these regions where uncertainties dominate, the estimates are expected to be relatively free of contamination (Morales et al. 2012). Within the instrumental window, results from Datta et al. (2010), Vedantham et al. (2012), Williams et al. (2012), and our study have shown that the wedge-shaped region in k-space is contaminated due to unsubtracted foreground sources and their sidelobes relatively more than in other regions in k-space. Hence, the EoR window for the H i power spectrum has been designated as the region in k-space inside the instrumental window excluding the wedge. The idea of an optimal window is investigated further.

We have shown that foregrounds are not strictly contained within the wedge-shaped region (see Section 5.3). A spillover from the wedge-shaped region is caused by the instrumental delay function $W_\eta ^{\rm B}(\eta)$. The characteristic width of this convolving instrumental response is proportional to $B_{\rm eff}^{-1}$. Thus, immediately following the wedge boundary determined by the horizon limit given by Equation (16), the spillover up to a few characteristic widths of convolution by the instrumental frequency response is also found to contain higher levels of contamination (see Figures 2, 5, and 6).

We investigate refinements to the so-called EoR window. We narrow the EoR window by adding a term proportional to the characteristic convolution width. We define a refined EoR window as the region

$$\begin{equation} k_\parallel \ge \frac{H_0\,E(z)\,D_{\rm M}(z)}{c\,(1+z)}\left(k_\perp + \frac{e}{B_{\rm eff}}\,\frac{2\pi \,f_{21}}{(1+z)\,D_{\rm M}(z)}\right). \end{equation} \tag{ 26 }$$

This will reduce to the horizon limit in Equation (16) without the second term in parentheses. The second term is proportional to the characteristic width of the convolution arising from instrumental delay function $W_\eta ^{\rm B}(\eta)$. e parameterizes this constant of proportionality. Equation (26) represents a family of EoR windows. This concept is illustrated in Figure 11. The instrumental window in k-space is shown as a gray box. The bottom right corner of this box is the region contaminated by foregrounds, and the top left corner represents the refined EoR window.

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8.2. One-dimensional Sensitivity

Using the model for the EoR H i power spectrum and with estimates of three primary uncertainties, namely, the foregrounds, thermal noise, and sample variance, the sensitivity of the instrument to EoR power spectrum detection may be obtained. By averaging signal and uncertainties in independent voxels in spherical shells of $k=(k_\perp ^2+k_\parallel ^2)^{1/2}$, sensitivity may be improved.

The model EoR H i power spectrum we have considered is spherically symmetric and hence a function only of radial coordinates in k-space. This symmetry is modified to an extent by the instrument observing the power spectrum because the instrumental term, $W_{uv\eta }(\overline{\mathbf {u}})$, in Equation (24) is not spherically symmetric. During spherical averaging, we ignore this loss of spherical symmetry caused by instrumental distortion. While averaging in shells of k, we average the observed signal in these shells and, correspondingly, estimate the uncertainties by adding them in inverse quadrature.

We compare the one-dimensional signal and noise estimates for the cases listed in Table 2, while deploying rectangular and extended Blackman–Nuttall band shapes, for a range of values of e.

Figures 12(a)–(d) show in detail the signal and uncertainties expected with the MWA for observing mode 1 listed in Table 2. Figure 12(a) demonstrates the levels of signal (solid circles) and uncertainty (solid line) expected with either of the bandpass windows employed for the EoR window parameter e = 0. Also shown are the individual components of the total uncertainty in different line styles (foregrounds: dot–dashed; thermal noise: dashed; sample variance: dotted). Figure 12(b) shows the change in the power spectrum of EoR H i and that of foregrounds when e is varied (e = 1, 2, 3 in red, green, and blue, respectively) relative to their respective values at e = 0 (black). Similarly, Figure 12(c) shows the change in thermal noise component of the power spectrum for different values of e, relative to its values at e = 0. In other words, Figures 12(b) and (c) show signal and uncertainty components normalized with respect to themselves obtained at e = 0. Hence, the quantities at e = 0 are shown for reference in these figures at a constant value of unity. Figure 12(d) shows the ratio of signal to uncertainty (S/N) as e is varied. The left sub-panel in each panel corresponds to a rectangular bandpass window, while that on the right is obtained with an extended Blackman–Nuttall window.

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With observing mode 1 using 128-tile MWA, the signal clearly appears to be detectable (S/N >1) for k ≲ 0.8 Mpc⁻¹ (see Figures 12(a) and (d)). From Figure 12(a), with e = 0, foreground contamination (dot–dashed line) exceeds thermal noise (dashed line) for k ≲ 0.2 Mpc⁻¹ and k ≲ 0.1 Mpc⁻¹ while using rectangular and Blackman–Nuttall windows, respectively. Beyond this crossover, thermal noise takes over as the dominant source of uncertainty in the power spectrum. Foregrounds (dot–dashed line) and sample variance (dotted line) are roughly equal up to this crossover. As e increases, progressively larger regions get excluded from k-space. This is clearly visible in Figures 12(b)–(d), especially for k ≲ 1 Mpc⁻¹, through a systematic drift of radii of spherical shells ("+" symbols) toward higher values as e increases (red to green to blue). In other words, increasing e from 0 to 3 makes it progressively harder to recover scales with 0.06 Mpc⁻¹ ≲ k ≲ 0.2 Mpc⁻¹. This results in partial removal of different uncertainties and the signal, besides an inherent decrease in signal strength with increasing k. However, the decrease in signal by ≲1–2 orders of magnitude (solid red, green, and blue curves) is less rapid than that in the foreground contamination (dashed red, green, and blue curves), which decreases by ≳1–2 orders of magnitude. This is true for both bandpass shapes but is quite pronounced for the extended Blackman–Nuttall window (see Figure 12(b)), where the foreground contamination reduces by ∼ 5–10 orders of magnitude. On the other hand, as e increases, the thermal noise component changes at most by a factor of two as seen from Figure 12(c). Effectively, the thermal noise component is only mildly affected compared to the signal and foregrounds. Regardless of the bandpass window used, the colored curves which are almost coincident in Figure 12(d) show that there is no improvement in overall sensitivity as e is varied. This is a consequence of the nature of inverse quadrature weighting used in averaging in spherical shells of k. However, it is very important to reiterate that the foreground contamination decreases more rapidly than the loss in signal as e is increased. In fact, foregrounds are almost completely removed for e ≳ 1 while using the Blackman–Nuttall window. Hence, using a combination of extended Blackman–Nuttall band shape with certain members of the family of EoR window (1 ≲ e ≲ 2) does not appear to improve MWA sensitivity, but offers significant leverage in reducing foreground contamination from the power spectrum, thereby providing a cleaner EoR window. This could become very significant when imperfect source subtraction (position and calibration errors) and extended emission from extragalactic and Galactic foregrounds are also taken into account. This is due to the 7–8 orders of magnitude of extra tolerance provided by an extended Blackman–Nuttall window relative to a rectangular window in the amount of spillover of foreground contamination into the EoR window.

We investigate the sensitivity for observing mode 2, where a total of 1000 hr were divided over 20 patches of sky to obtain 20 independent measurements of power spectrum. This differs from observing mode 1 in that the total observing time is now divided on multiple fields. Figure 13(a) is the counterpart of Figure 12(a) and illustrates the signal and different uncertainty components obtained with the MWA in this observing mode. Counterparts to Figures 12(b) and (c) will be identical and are not shown. Figure 13(b) shows the ratio of signal to total uncertainty (S/N) for this observing mode.

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Since the time on individual fields has reduced by a factor of N_fields = 20, leading to a reduction in the number of coherent visibility measurements per field, the thermal noise component in each measurement of the power spectrum has increased by the same factor. The foreground contamination and sample variance in an individual power spectrum measurement remain identical to those in observing mode 1, where only a single field is observed. When independent measurements of power spectra are averaged, all the components of uncertainty in the averaged power spectrum are reduced by a factor $\sqrt{N_{\rm fields}}\approx 4.47$ relative to what they were in the individual measurements of power spectra. The net result, relative to observing mode 1, is that the foreground contamination and sample variance have reduced by a factor 4.47, while the thermal noise component has worsened by the same amount. This is evident when Figure 13(a) is compared with Figure 12(a). As a result, the crossover, up to which foreground contamination and sample variance dominate over thermal noise, moves leftward and now occurs at k ≃ 0.1 Mpc⁻¹ for both bandpass windows. This has a twofold effect on overall sensitivity relative to that in the previous case: (1) the sensitivity in lowest bin of k (k ≃ 0.06 Mpc⁻¹), where sample variance and foreground components dominated over the thermal noise component in observing mode 1, has improved to 20, a factor of ≈4 (consistent with $\sqrt{N_{\rm fields}}\approx 4.47$); and (2) the sensitivity in other bins of k (k ≳ 0.1 Mpc⁻¹) has degraded because the thermal noise component, which was already dominant in this regime, has worsened. The final effect on detectability is that S/N >1 only for k ≲ 0.4 Mpc⁻¹.

How does sensitivity in observing mode 2 compare to that in observing mode 1? Sensitivity is the result of a complex interplay between the relative magnitudes of the desired EoR H i signal and different uncertainty components in the power spectrum. As far as MWA is concerned, the thermal noise component appears to be dominant on scales with k ≳ 0.1–0.2 Mpc⁻¹. Hence, dividing the observing time over multiple fields and averaging the independent measurements of power spectra helps improve the sensitivity by a factor of ≈4 for k ≲ 0.1 Mpc⁻¹ but degrades it everywhere else. Consequently, the zone of detectability (S/N >1) in k-space becomes narrower from k ≲ 0.8 Mpc⁻¹ to k ≲ 0.4 Mpc⁻¹. Thus, except for an improvement in sensitivity on the largest scales (k ≲ 0.1 Mpc⁻¹), increasing the number of independent fields seems to offer no significant advantage.

As far as effects of bandpass windowing and family of EoR windows are concerned, significant improvement in MWA sensitivity is seen neither with bandpass window shapes nor with the EoR window parameter e. However, it is crucial to obtain the cleanest EoR window possible in order to reduce the amount of systematics in the data, which may not be fully understood, such as those arising from unsubtracted foreground residuals. We find that refinements to the EoR window (through parameter e), a Blackman–Nuttall bandpass window shape, and a combination of both significantly reduce the extragalactic foreground contamination in the measured power spectrum.

The primary goal of this work was to understand and estimate some of the fundamental factors that limit sensitivity of an EoR H i power spectrum measurement, namely, point-like extragalactic foreground contamination, thermal noise, and sample variance of the H i brightness temperature fluctuations. A secondary goal was to understand how these uncertainties compete with each other on different scales in determining the sensitivity of a radio interferometer array, such as the MWA, toward detection of the EoR H i power spectrum.

An analytic-cum-statistical approach was used in representing residual image cubes by assuming a radio source count distribution on the sky. For the 128-tile MWA at 170.7 MHz, when sources above the 5σ classical source confusion threshold were subtracted, the 1σ classical source confusion limit near the zenith was found to be ≈35 mJy for a natural weighting scheme.

Unsubtracted foreground sources and their sidelobes contaminate the predicted EoR signal. The frequency dependence of synthesized beam distributes the contamination from sidelobes onto a wedge-shaped region in k-space. We have presented a unified framework for signal and noise estimation, using which we estimate foreground contamination in three-dimensional k-space. This framework also attempts to take into account multi-baseline mode-mixing effects caused by loss of coherence between non-identical baselines inside an independent cell in the spatial frequency domain. Using this framework, we establish an expression for the boundary of the wedge set by the horizon limit.

We show for the first time, quantitatively, how the usage of a finite bandpass spills the contamination from unsubtracted sources and their sidelobes into the EoR window. This spillover decreases by 7–8 orders of magnitude, for instance, in the range 0.2 Mpc⁻¹ ≲ k_∥ ≲ 5 Mpc⁻¹ at k_⊥ ≈ 0.01 Mpc⁻¹, by switching from a rectangular to an extended Blackman–Nuttall window. We argue that this additional tolerance provided by the latter could prove to be of crucial significance in minimizing power spectrum contamination when the impact of imperfect source subtraction (due to position and calibration errors) and extended extragalactic and Galactic foregrounds are also considered. The frequency weighting in an extended Blackman–Nuttall bandpass window also lowers the thermal noise component of the power spectrum by 28% relative to that achievable with a rectangular window. Conversely, in order to achieve the same thermal noise power with both windows, the duration of observing with a rectangular window has to be ≈40% longer when compared to an extended Blackman–Nuttall window.

We performed case studies of two different observing modes—6 hr synthesis repeated for a total of 1000 hr on a single field, and 6 hr synthesis repeated on 20 independent fields for a total of 1000 hr—and studied the effects of the aforementioned uncertainties on EoR H i power spectrum detection using the MWA. In both cases, detection appears to be possible (S/N >1). A total of 1000 hr on a single field shows that the signal is detectable on scales with k ≲ 0.8 Mpc⁻¹, while dividing it over 20 independent fields narrows the zone of detectability to scales with k ≲ 0.4 Mpc⁻¹. Since sample variance and foregrounds, rather than thermal noise, are the dominant uncertainties on the largest scales (k ≲ 0.1 Mpc⁻¹), detection sensitivity on these scales improves by roughly four times if the observing time is divided over 20 independent regions of sky.

The concept of EoR window was probed quantitatively. Foreground contamination can be drastically reduced by using an extended Blackman–Nuttall bandpass window and through refinements (1 ≲ e ≲ 2) to the EoR window.

By modeling in detail the various uncertainties, we have shown the significance of different uncertainties on various scales and their roles in determining overall sensitivity. Observing many independent fields of view worsens the thermal noise and hence degrades the sensitivity for the MWA relative to a single field observation of the same duration. Bandpass window shaping and refinements to the EoR window do not affect sensitivity, but have a significant effect on containing the foreground contamination.

This scientific work makes use of the Murchison Radio-astronomy Observatory. We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site. Support for the MWA comes from the U.S. National Science Foundation (grants AST-0457585, PHY-0835713, CAREER-0847753, and AST-0908884), the Australian Research Council (LIEF grants LE0775621 and LE0882938), the U.S. Air Force Office of Scientific Research (grant FA9550-0510247), and the Centre for All-sky Astrophysics (an Australian Research Council Centre of Excellence funded by grant CE110001020). Support is also provided by the Smithsonian Astrophysical Observatory, the MIT School of Science, the Raman Research Institute, the Australian National University, and the Victoria University of Wellington (via grant MED-E1799 from the New Zealand Ministry of Economic Development and an IBM Shared University Research Grant). The Australian Federal government provides additional support via the National Collaborative Research Infrastructure Strategy, Education Investment Fund, and the Australia India Strategic Research Fund, and Astronomy Australia Limited, under contract to Curtin University. We acknowledge the iVEC Petabyte Data Store, the Initiative in Innovative Computing and the CUDA Center for Excellence sponsored by NVIDIA at Harvard University, and the International Centre for Radio Astronomy Research (ICRAR), a Joint Venture of Curtin University and the University of Western Australia, funded by the Western Australian State government. The National Radio Astronomy Observatory is a facility of the National Science Foundation operated under cooperative agreement by Associated Universities, Inc.

At any given frequency, the field of view contains many sources. Any given synthesized beam area of an array consists of many unresolved sources along that line of sight. Due to the statistical nature of the distribution of sources, the flux density contained in a synthesized beam area varies across the sky. The classical source confusion is the variation in flux density due to random distribution of unresolved sources across different beam areas on the sky. The theory on confusion is discussed in detail in Condon (1974) and Rohlfs & Wilson (2000). This paper uses the discussion and notations presented in the latter.

The differential number density of sources per unit solid angle with respect to flux density is denoted by dn/dS, where dn is the number of sources per steradian in a flux density interval between S and S + dS. The variance in confusing source flux density in a given solid angle, Ω, due to the number count distribution of sources is given by

$$\begin{equation} \sigma _{\rm C}^2=\Omega \int _{S_{\rm min}}^{S_{\rm C}} S^2\,\frac{{d}n}{{d}S}\,{d}S, \end{equation} \tag{ A1 }$$

where S_min and S_C are the lower and upper limits on the flux density of radio sources, respectively.

Since $\sigma _{\rm C}^2$ is determined from the entire range of flux densities up to S_C, if there are bright sources, $\sigma _{\rm C}^2$ will be overestimated. Hence, we perform an iterative procedure wherein all foreground sources brighter than ρ_C σ_C and their sidelobes are subtracted, thereby updating the upper end of the flux density range, S_C. Here, ρ_C acts as the source subtraction threshold factor. This iterative procedure of subtracting foreground sources brighter than S_C = ρ_C σ_C, and updating S_C and σ_C, is performed until there are no sources brighter than ρ_C σ_C. We consider the $\sigma _{\rm C}^2$ so computed as the "true" classical source confusion variance.

In practice, usually, deconvolution procedures can estimate and subtract foreground sources and their associated sidelobes to a limited extent, leaving behind a residual image. With the knowledge of distribution of radio sources encoded in dn/dS, a set threshold factor (ρ_C), and the solid angle (Ω) corresponding to the angular resolution limit, the depth of source subtraction and corresponding residuals in the residual image can be determined using the iterative procedure described above and is given by the following equation (Rohlfs & Wilson 2000):

$$\begin{equation} \rho _{\rm C}^2 = \frac{S_{\rm C}^2}{\Omega \int _{S_{\rm min}}^{S_{\rm C}} S^2\,\frac{{d}n}{{d}S}\,{d}S}. \end{equation} \tag{ A2 }$$

This is obtained by imposing the criterion that the residual image is allowed to have unsubtracted sources of flux densities up to S_C ⩽ ρ_C σ_C. We have assumed that the source subtraction threshold, S_C, is set only by the classical source confusion noise, whereas, in practice, confusion caused by sidelobes in a residual image will also play a role in determining the source subtraction threshold in this iterative procedure.

We use the radio source statistics provided by Hopkins et al. (2003). Their best-fit expression for the source counts is

$$\begin{equation} \log [({d}n/{d}S)/(S^{-2.5})] = \sum _{i=0}^6 a_i \,[\log (S/{\rm mJy})]^i, \end{equation} \tag{ A3 }$$

valid at 1400 MHz for 0.05 mJy ⩽ S ⩽ 1000 mJy, where a₀ = 0.859, a₁ = 0.508, a₂ = 0.376, a₃ = −0.049, a₄ = −0.121, a₅ = 0.057, and a₆ = −0.008. S is in units of mJy, and (dn/dS)/S^−2.5 is in units of Jy^1.5 sr⁻¹. The normalization by S^−2.5 indicates that the expression is relative to a Euclidean universe.

The solid angle at various locations of an image in (l, m)-coordinates scales as

$$\begin{eqnarray} \Omega &= \frac{\Delta l\,\Delta m}{\sqrt{1-l^2-m^2}}, \end{eqnarray} \tag{ A4 }$$

where Δl Δm is the pixel size. This implies, from Equation (A2), that confusion variance (σ_C) and flux density cutoff (S_C) are functions of position in the residual image. Estimating confusion noise over a wide range of solid angles requires an extrapolation of the empirical function in Equation (A3) on both ends of the flux density range. We have extrapolated at the higher end by a flat function mimicking a Euclidean local universe behavior, and at the lower end by an extension of the same slope found between 2 and 20 mJy. Figure 14(a) shows Equation (A3) extrapolated at both ends beyond the aforementioned range in flux density (specified by vertical dotted lines). The dashed segments S < 0.05 mJy and 2 mJy ⩽ S ⩽ 20 mJy have slopes identical to each other.

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Under the assumption that the source population remains the same at the relevant frequency (170.7 MHz, for instance) under consideration as at 1400 MHz, and that these sources are unresolved with the MWA, the same expression for the distribution of source counts can be used once the spectral index is taken into account. There has been conflicting evidence in literature (Randall et al. 2012, and references therein) over the spectral index properties of radio sources at frequencies below 1.4 GHz and whether the spectral index flattens for faint sources at low frequencies. Further, Kellermann (1964) has pointed out that spectral index distribution is not independent of the observing frequency since sources with flatter spectral index are more likely to be observed at higher frequencies and those at lower frequencies tend to have steeper spectral index, and has subsequently provided a spectral index correction that tends to offset this bias. However, for our work, we have adopted a mean spectral index for radio sources as α = −0.78 (Ishwara-Chandra et al. 2010), where S ∝f^α. Compared to flatter values of spectral index, our adopted value of α = −0.78 gives us higher values for the classical source confusion at 170.7 MHz, making our sensitivity estimates conservative.

Sources can be subtracted down to various levels of threshold factor, ρ_C. Figure 14(b) represents a numerical solution for Equation (A2) using dn/dS from Equation (A3) for various values of ρ_C, as a function of the beam solid angle, Ω. As expected from Equation (A2), Figure 14(b) shows that σ_C and S_C are nonlinear functions of both ρ_C and Ω. σ_C depends on the choice of ρ_C: it increases with ρ_C.

Following the notations established in Section 3, the true visibilities of foregrounds, $V_{uvf}^{\rm FG;T}(\underline{\mathbf {u}},f)$, are modified by the instrument as

$$\begin{eqnarray} V_{uv\eta }^{\rm FG}(\overline{\mathbf {u}}) &= \int V_{uvf}^{\rm FG;T}(\underline{\mathbf {u}},f)\,S_{uv}(\underline{\mathbf {u}})\,W_f^{\rm B}(f) \ast W_{uv}^{\rm P}(\underline{\mathbf {u}})\,e^{-j2\pi \eta f}{d}f \nonumber \\ &= V_{uv\eta }^{\rm FG;T}(\overline{\mathbf {u}})\,S_{uv}(\underline{\mathbf {u}}) \ast W_\eta ^{\rm B}(\eta) \ast W_{uv}^{\rm P}(\underline{\mathbf {u}}) \nonumber \\ &= V_{uv\eta }^{\rm FG;T}(\overline{\mathbf {u}})\,S_{uv}(\underline{\mathbf {u}}) \ast W_{uv\eta }(\overline{\mathbf {u}}), \end{eqnarray} \tag{ B1 }$$

where the true foreground visibilities, $V_{uvf}^{\rm FG;T}(\underline{\mathbf {u}},f)$, have been convolved with the spatial frequency response of the antenna's power pattern, $W_{uv}^{\rm P}(\underline{\mathbf {u}})$, multiplied by the sampling function, $S_{uv}(\underline{\mathbf {u}})$, in the (u, v)-plane, multiplied by the bandpass window function in frequency, $W_f^{\rm B}(f)$, and Fourier transformed along frequency to obtain the measurements in Fourier space ($\overline{\mathbf {u}}$). $W_\eta ^{\rm B}(\eta)$ is the Fourier transform of $W_f^{\rm B}(f)$, and $j=\sqrt{-1}$. $W_{uv\eta }(\overline{\mathbf {u}})=W_{uv\eta }^{\rm P}(\overline{\mathbf {u}})\ast W_\eta ^{\rm B}(\eta)$, where $W_{uv\eta }^{\rm P}(\overline{\mathbf {u}})$ may be interpreted as the spatial frequency response of the antenna's power pattern over an infinitely uniform bandpass. Assuming that changes in the antenna power pattern over the observing band are insignificant, $W_{uv}^{\rm P}(\underline{\mathbf {u}})=W_{uv\eta }^{\rm P}(\underline{\mathbf {u}},\eta =0)$.

The covariance matrix for the measured foreground visibilities in Fourier space may be written as

$$\begin{eqnarray} C^{\rm FG}(\overline{\mathbf {k}}_i,\overline{\mathbf {k}}_j) &=& \left \langle V_{uv\eta }^{\rm FG}(\overline{\mathbf {u}}_i)^\star V_{uv\eta }^{\rm FG}(\overline{\mathbf {u}}_j)\right \rangle \nonumber \\ &=& \int \int \int \,\int \int \int \left \langle V_{uv\eta }^{\rm FG;T}(\overline{\mathbf {u}}_p)^\star V_{uv\eta }^{\rm FG;T}(\overline{\mathbf {u}}_q) \right \rangle \nonumber\\ &&\times S_{uv}(\underline{\mathbf {u}}_p)^\star \,S_{uv}(\underline{\mathbf {u}}_q)\,W_{uv\eta }(\overline{\mathbf {u}}_i-\overline{\mathbf {u}}_p)^\star \nonumber\\ &&\times W_{uv\eta } (\overline{\mathbf {u}}_j-\overline{\mathbf {u}}_q)\,{d}^3 \overline{\mathbf {u}}_p\, {d}^3 \overline{\mathbf {u}}_q, \end{eqnarray} \tag{ B2 }$$

where ⋆ denotes a complex conjugate. But, being an uncorrelated statistical signal, $\langle V_{uv\eta }^{\rm FG;T}(\overline{\mathbf {u}}_p)^\star V_{uv\eta }^{\rm FG;T}(\overline{\mathbf {u}}_q) \rangle = P^{\rm FG}(\overline{\mathbf {u}}_p)\,\delta (\overline{\mathbf {u}}_p-\overline{\mathbf {u}}_q)$. Hence,

$$\begin{eqnarray} C^{\rm FG}(\overline{\mathbf {k}}_i,\overline{\mathbf {k}}_j) &=& \int \int \int P^{\rm FG}(\overline{\mathbf {u}})\,|S_{uv}(\underline{\mathbf {u}})|^2\,W_{uv\eta }(\overline{\mathbf {u}}_i-\overline{\mathbf {u}})^\star \nonumber\\ &&\times W_{uv\eta } (\overline{\mathbf {u}}_j-\overline{\mathbf {u}})\,{d}^3 \overline{\mathbf {u}} \nonumber \\ &=& \int \int \int P^{\rm FG}(\overline{\mathbf {u}}) |S_{uv}(\underline{\mathbf {u}})|^2\,W_{uv}^{\rm P}(\underline{\mathbf {u}}_i-\underline{\mathbf {u}})^\star \nonumber\\ &&\times W_{uv}^{\rm P}(\underline{\mathbf {u}}_j-\underline{\mathbf {u}})\,W_\eta ^{\rm B}(\eta _i-\eta)^\star \,W_\eta ^{\rm B}(\eta _j-\eta)\,{d}^3 \overline{\mathbf {u}}. \nonumber\\ \end{eqnarray} \tag{ B3 }$$

The power spectrum is simply the diagonal of the covariance matrix, i.e., when the intensities at locations are compared with themselves: $P^{\rm FG}_{\rm inst}(\overline{\mathbf {k}}) = C^{\rm FG}(\overline{\mathbf {k}}_i,\overline{\mathbf {k}}_j)\,\delta _{ij}$. Thus,

$$\begin{eqnarray} &&P^{\rm FG}_{\rm inst}(\overline{\mathbf {u}}) = \int \int \int P^{\rm FG}(\overline{\mathbf {u^\prime }})|S_{uv}(\underline{\mathbf {u^\prime }})|^2\, |W_{uv\eta }(\overline{\mathbf {u}}-\overline{\mathbf {u^\prime }})|^2\,{d}^3 \overline{\mathbf {u^\prime }}. \nonumber\\ \end{eqnarray} \tag{ B4 }$$

An insight into mode-mixing may be obtained if the above expression for the power spectrum is re-expressed as a Fourier transform of quantities in the (l, m, f)-coordinates as follows:

$$\begin{eqnarray} P^{\rm FG}_{\rm inst}(\overline{\mathbf {u}}) &=& \int \int \int P_{lmf}^{\rm FG}(\overline{\mathbf {l}})\,\ast \,|S_{lm}(\underline{\mathbf {l}})|^2 \, \left|W_{lm}^{\rm P}(\underline{\mathbf {l}})\right|^2 \, \left|W_f^{\rm B}(f)\right|^2\nonumber\\ &&\times e^{-j2\pi (\underline{\mathbf {u}}\cdot \underline{\mathbf {l}}+\eta f)}\,{d}^3 \, \overline{\mathbf {l}}. \end{eqnarray} \tag{ B5 }$$

Confusion variance, $P_{lmf}^{\rm FG}(\overline{\mathbf {l}})$, from extragalactic foreground sources will be considered as the cause for foreground contamination in the power spectrum. It may be computed using Equations (A2) and (A4). When line-of-sight (f and η) terms are dropped, the integrand in the above equation is consistent with that in Equation (24) of Bowman et al. (2009).

We further assume that $P_{lmf}^{\rm FG}$ is independent of frequency, provided that the residuals have no spectral variations. Noting that $\underline{\mathbf {u}}=f\,\underline{\mathbf {x}}/c$, the above equation may be rewritten as

$$\begin{eqnarray} P^{\rm FG}_{\rm inst}(\overline{\mathbf {u}}) &=& \int \int \int P_{lm}^{\rm FG}(\underline{\mathbf {l}})\,\ast \,|S_{lm}(\underline{\mathbf {l}})|^2\, \left|W_{lm}^{\rm P}(\underline{\mathbf {l}}) \right|^2\, \left|W_f^{\rm B}(f) \right|^2 \nonumber\\ &&\times e^{-j2\pi (\frac{\underline{\mathbf {x}}\cdot \underline{\mathbf {l}}}{c}+\eta)f}\,{d}^2 \underline{\mathbf {l}}\;{d}f \nonumber \\ &=& \int \int P_{lm}^{\rm FG}(\underline{\mathbf {l}})\,\ast \,|S_{lm}(\underline{\mathbf {l}})|^2\,\left|W_{lm}^{\rm P}(\underline{\mathbf {l}}) \right|^2\,\ast \,\left|W_\eta ^{\rm B}(\eta) \right|^2 \nonumber\\ &&\ast \delta \left(\eta +\frac{\underline{\mathbf {x}}\cdot \underline{\mathbf {l}}}{c}\right)\,{d}^2 \underline{\mathbf {l}}, \end{eqnarray} \tag{ B6 }$$

where $\underline{\mathbf {x}}$ is the baseline vector in units of distance. The argument of the delta function connects transverse spatial structure to that along the line of sight. This demonstrates the mode-mixing aspect of contamination from foregrounds in the power spectrum.