REDUNDANT ARRAY CONFIGURATIONS FOR 21 cm COSMOLOGY

Although a 331-element hexagonal core measures 54,615 visibilities at any one time, it actually only measures 630 unique separations between antennas. These fall on a hexagonal grid, since the convolution of a hexagon with itself is simply a larger hexagon. In a compact array, the minimum separation between antennas, 14.6 m, sets the minimum baseline length and the fundamental size of that grid of measured baselines. And since the Fourier sky is regularly sampled at spacings much wider than Nyquist sampling (λ/2), we observe grating lobes in the PSF due to signal aliasing caused by undersampling. If we want to suppress grating lobes, we need denser sampling and more independent information.

Of course, our visibility measurements are not disconnected δ-functions in Fourier space. The visibility, which is a measurement of the correlation of voltages measured by two antennas separated by baseline vector ${\boldsymbol{b}}$, is given by

$$\begin{eqnarray}&&V({\boldsymbol{b}},\nu )=\int B(\hat{{\boldsymbol{s}}},\nu )I(\hat{{\boldsymbol{s}}},\nu )\exp \left[-2\pi i\displaystyle \frac{\nu }{c}{\boldsymbol{b}}\cdot \hat{{\boldsymbol{s}}}\right]d{\rm{\Omega }}.\end{eqnarray} \tag{ 1 }$$

where I($\hat{{\boldsymbol{s}}}$, ν) is the sky intensity as a function of position and frequency and B($\hat{{\boldsymbol{s}}}$, ν) is the product of the direction and frequency dependence of the response of individual antennas.⁵

For a coplanar array sampling the sky instantaneously, we can think of a visibility as a average over a region of the Fourier transformed sky, projected onto a plane and convolved with the Fourier transform of the beam product (Thompson et al. 2001). Alternatively, we can think of it as a δ-function sampling of the beam-convolved sky. HERA's beams are roughly axisymmetric and therefore our weighted average is over an approximately circular region of Fourier space (also known as the uv-plane). It follows that we have the least information about the parts the uv-plane furthest from the exact baselines probed. If we would like to make new measurements inside the core (in uv-space) as independent as possible (although not, of course, completely independent of existing baselines), then we would like to place antennas so as to produce new baselines at the centroids of the lattice of equilateral triangles that make up the hexagonal grid.

For the sake of comparison, we start with a "fiducial" HERA core, which we call configuration (a) and show in the first panel of Figure 1. In Figure 2 we can see that hexagonal grid pattern of instantaneously redundant baselines (as opposed to baselines that measure the same mode at a later time due to the rotation of the Earth.) We note that the best-sampled baselines are the shortest. Longer separations have fewer corresponding pairs of antennas within the core.

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To produce new baselines at the centroids of the triangular lattice, we need to add antennas that are not on the hexagonal grid or dither existing antennas. If we pick any two 14.6 m basis vectors that define the hexagonal grid in linear, integer combinations (e.g., up-and-to-the-left and up-and-to-the-right), we note the centroids fall at 1/3 or 2/3 of the sum of those two vectors. In other words, by introducing antennas dithered from the main grid by 1/3 or 2/3 of that separation, we can add a grid of baselines that effectively subsample the main hexagonal grid. Maximally packed dishes Nyquist sample modes within the main lobe of the primary beam, but they produce grating lobes in the sidelobes (which would require λ/2 spacing to eliminate for the whole sky). Sub-aperture sampling suppresses these grating lobes, as we will see in Section 3.

We employ this strategy in developing our two alternative array configurations, (b) and (c), shown in Figure 1. In (b), we add a small number of antennas near the perimeter of the main hexagon, each displaced from the main grid. In (c) we split the entire hexagon into three sectors, each relatively displaced by those same increments. Both patterns allow for complete sampling at triple the density of the original grid (see Figure 2). In the case of the split-core configuration (c), that triple-density coverage does not extend to the full hexagonal core, but those baselines can be filled in by outrigger antennas, as we will discuss in Section 2.2.⁶

An important contrast between configurations (b) and (c) is that while off-grid baselines in (b) have at most a handful of measurements, the distinction between "on-grid" and "off-grid" in (c) becomes less clear for longer baselines. That is because longer baselines are more likely to be inter-sector and thus off-grid while shorter baselines are more often intra-sector and thus on-grid. The result is more distributed and thus uniform coverage of the triple-density grid than in (b).⁷ It should be noted that configurations (a) and (b), by virtue of their denser cores, concentrate somewhat more sensitivity on shorter baselines, which are those less affected by the wedge. We also note that (b) and (c) do not substantially affect the possibility of performing FFT correlation, since it merely requires that all antennas fall on a regular or hierarchically regular grid (Tegmark & Zaldarriaga 2010); gridding based FFT schemes (Morales 2011; Thyagarajan et al. 2015a) are not necessary. That grid is now simply finer and has many unobserved vertices that must be flagged.

In addition to the hexagonal cores, we are interested in supplementing each design with several far-flung "outrigger" antennas in order to improve angular resolution (see Figure 3). Adding a single antenna far from the core adds an entire hexagon of baselines to the uv-plane. This improves the imaging capability of the array and can help with foreground removal (see Section 3). With configuration (a), we pick the minimal set of additional antennas to tile the maximum area of the uv-plane with hexagons. Although hexagons do tessellate, the borders between them create discontinuities in the hexagonal grid, as we can see in Figure 4. Tessellating hexagons produces some overlapping regions in the core, but outside the core the scheme is extensible to longer baselines with no additional overlap.

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Configuration (b) takes inspiration from configuration (a) by also using the hexagonal perimeter of the core to tile the uv-plane. Its outrigger antennas differ in two notable ways. First, it includes outrigger antennas on both sides of the core. Second, that grid is contracted slightly from the grid in (a) such that there exist redundant baselines measured between different outrigger antennas. Both these changes are designed to allow for redundant calibratability, which we will discuss in Section 4.

Finally, configuration (c) takes a different approach. While it too is designed for redundant calibratability, it takes advantage of the split-core configuration to create full baseline coverage at triple density (see Figure 4). Compared to (a) or (b), that increased density allows it to better suppress the sidelobes of point sources at the cost of some angular resolution.

Before we move on to compare these configurations in terms of mapmaking and calibration, it is useful to check that these designs do not differ significantly in terms of their raw sensitivity to the 21 cm power spectrum. Since most sensitivity comes from short baselines and all three configurations have very similar cores, we do not expect much variation.

A proper calculation of the significance with which a fiducial model of reionization can be detected by these arrays is quite difficult. In addition to depending on foreground removal strategies (Pober et al. 2014) and models for the covariance of foreground residuals (Dillon et al. 2015a), one should take into account the partial coherence of baselines that measure overlapping modes in the uv-plane, either from the same baseline at different times or from sub-aperture sampling. This is possible, although computationally challenging. No algorithm exists yet that propagates these effects into the power spectrum (Dillon et al. 2015b). A full treatment is therefore outside the scope of this paper.

Instead, we follow Pober (2015) and employ the 21cmSense code⁸ to explore the sensitivities of three configurations. This is quantified in terms of a "cumulative detection significance," which is the square root of the sum in quadrature of the ratio of the 21 cm power spectrum to the thermal noise over all wavenumbers k. In particular, we look at an 8 MHz bandwidth centered on 135 MHz for a 1080 hr drift scan and calculate the sensitivity to a model power spectrum from Mesinger et al. (2011) with a midpoint of reionization at z = 9.5. Table 1 shows that, regardless of our foreground mitigation strategy, all three of our arrays perform very similarly. We therefore should consider other criteria to assess their relative merit.

Fundamentally the problem of mapmaking is one of data reduction. We have a large quantity of time-ordered measurements, expressed as a data vector ${\boldsymbol{y}}$, which sample different linear combinations of the sky at different times and in different ways. However, those data are noisy, so we must be cognizant of how we weight and combine them.

The action of the interferometer measuring visibilities (as in Equation (1)) can be expressed as a relationship between some discretized sky ${\boldsymbol{x}}$ and our measurements ${\boldsymbol{y}}$ as

$$\begin{eqnarray}&&{\boldsymbol{y}}={\boldsymbol{Ax}}\,+{\boldsymbol{n}}.\end{eqnarray} \tag{ 2 }$$

Here ${\boldsymbol{A}}$ represents the interferometric measurement—the family of integrals like Equation (1) but discretized—and ${\boldsymbol{n}}$ is the thermal noise. The discretization of ${\boldsymbol{A}}$ is demonstrated in Dillon et al. (2015b), which also includes a more detailed discussion of the following definitions and derivations.

If the statistics of the thermal noise are Gaussian and we define ${\boldsymbol{N}}\equiv \langle {\boldsymbol{n}}\,{{\boldsymbol{n}}}^{\dagger }\rangle$, then the optimal estimator (in the sense that no information about the sky is lost), $\widehat{{\boldsymbol{x}}},$ is

$$\begin{eqnarray}&&\widehat{{\boldsymbol{x}}}={{\boldsymbol{DA}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{y}}\end{eqnarray} \tag{ 3 }$$

where ${\boldsymbol{D}}$ is some invertible normalization matrix (Tegmark 1997). To understand the statistics of this estimator, we should look at its mean and covariance. Since the sky is not random and since the thermal noise has $\langle {\boldsymbol{n}}\rangle =0$,

$$\begin{eqnarray}\langle \widehat{{\boldsymbol{x}}}\rangle & = & \langle {\boldsymbol{D}}\,{{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}({\boldsymbol{Ax}}\,+{\boldsymbol{n}})\rangle \\ & = & {{\boldsymbol{DA}}}^{\dagger }{{\boldsymbol{N}}}^{-1}({\boldsymbol{Ax}}\,+\langle {\boldsymbol{n}}\rangle )\\ & = & {{\boldsymbol{DA}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{Ax}}={\boldsymbol{Px}},\end{eqnarray} \tag{ 4 }$$

where we have defined

$$\begin{eqnarray}&&{\boldsymbol{P}}\equiv {\boldsymbol{D}}\,{{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}.\end{eqnarray} \tag{ 5 }$$

The matrix ${\boldsymbol{P}}$ is the matrix of PSFs that tells us how a source in any pixel in ${\boldsymbol{x}}$ will appear in all other pixels. Likewise the covariance of the estimator is given by

$$\begin{eqnarray}{\boldsymbol{C}} & = & \langle (\widehat{{\boldsymbol{x}}}-\langle \widehat{{\boldsymbol{x}}}\rangle ){(\widehat{{\boldsymbol{x}}}-\langle \widehat{{\boldsymbol{x}}}\rangle )}^{\dagger }\rangle \\ & = & \langle ({{\boldsymbol{DA}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{n}}){({{\boldsymbol{DA}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{n}})}^{\dagger }\rangle \\ & = & {{\boldsymbol{DA}}}^{\dagger }{{\boldsymbol{N}}}^{-1}\langle {\boldsymbol{n}}\,{{\boldsymbol{n}}}^{\dagger }\rangle {{\boldsymbol{N}}}^{-1}{{\boldsymbol{AD}}}^{\dagger }\\ & = & {{\boldsymbol{PD}}}^{\dagger }.\end{eqnarray} \tag{ 6 }$$

Since the only random thing here is the noise, this ${\boldsymbol{C}}$ is really the noise covariance of every map pixel with every other map pixel.

For estimating power spectra, Dillon et al. (2015b) argued that choosing a simple diagonal form of ${\boldsymbol{D}}$ is ideal for propagating mapmaking statistics into the power spectrum. In this work, we simply take ${\boldsymbol{D}}$ ∝ ${\boldsymbol{I}}$ such that the PSF of a source at zenith peaks at 1. In the flat-sky approximation used by most radio interferometers with small fields of view, this choice is analogous to natural weighting, where every independent measurement is weighted only by the noise in that measurement and not by the number of other measurements that fall in the same uv-cell.

However, it is also useful to consider the other extreme choice for ${\boldsymbol{D}}$. If we wanted an unbiased estimator of the sky, one where $\langle \widehat{{\boldsymbol{x}}}\rangle ={\boldsymbol{x}}$, then we should choose ${\boldsymbol{D}}$ = (${{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}$)⁻¹ in order to obtain ${\boldsymbol{P}}$ = ${\boldsymbol{I}}$. In the flat-sky approximation, this choice is analogous to uniform weighting of cells in the uv-plane. With a limited field of view (low uv-resolution) and limited angular resolution (small maximum u and v), it is possible to sample every uv-cell and produce a δ-function PSF using uniform weighting. This generally produces very noisy maps. And of course, depending on the pixelization, ${{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}$ is often not invertible and this procedure becomes impossible without further assumptions about the sky. However, when this procedure is possible, it follows then that

$$\begin{eqnarray}&&{\boldsymbol{C}}={{\boldsymbol{PD}}}^{\dagger }\longrightarrow {({{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}})}^{-1}.\end{eqnarray} \tag{ 7 }$$

Since inverse covariance weighting for a future data reduction step involves multiplication by ${\boldsymbol{C}}$⁻¹ = ${{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}$, this means that ${{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}$ measures the information content in our maps. In Section 3.3 we will explore the structure of this matrix for our array configurations.

We start with a more qualitative comparison of the PSFs of the three configurations. In Figure 5 we examine the central region of a high-resolution HEALPix (Górski et al. 2005) map of the PSF of a source in the center of a 20° facet (N_side = 256). At 150 MHz, that 20° corresponds to roughly double the size of HERA's primary beam. We examine the PSF produced by a single 10 s snapshot with the facet center at zenith and by a 40 minute drift scan observation centered on the facet. We also examine two weighting schemes, one where ${\boldsymbol{N}}$ simply reflects noise on a given redundantly sampled baseline (e.g., "natural" weighting) and another where no redundant baseline is weighted more than 50 times higher than any other. This is akin to "robust" weighting (Thompson et al. 2001), which transitions from natural weighting of low signal-to-noise ratio (S/N) baselines to uniform weighting of high S/N baselines, balancing noise against sidelobe confusion.

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At first glance, the PSFs are very similar. This is to be expected, especially for the naturally weighted PSFs, which are so dominated by the core baselines. The prominent six-fold sidelobes we see in all panels of Figure 5 are due to the inter-element spacing and configuration in the core.

There are some small differences between the configurations. The main lobe of the PSF is slightly more sharply defined in configurations (a) and (b) because their uv-coverage extends to longer baselines (see Figure 4). More prominently, the sidelobes are dimmer and more smeared out in configuration (c) than in either (a) or (b). This makes sense since we expected the sidelobes to be suppressed in the split-core configuration, which gives more weight to the off-grid baselines. Our "robust" weighting scheme helps us see this more clearly, since it gives relatively more weight to off-grid baselines and better demonstrates the sidelobe suppression possible with configuration (c).

The observed sidelobe suppression in Figure 5 raises a question: does configuration (c) actually have an improved ability to measure modes far from the center of the primary beam "more independently"? To answer that, we need to analyze the structure of the matrix ${{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}$, the noise-weighted instrument response which contains the information and correlations in our map. Were it computationally feasible, we would do this for a full-sky, high-resolution map. However, the numerical difficulty of applying the rigorous mapmaking treatment of Section 3.1 necessitates some approximations (Dillon et al. 2015b).

3.3.1. Widefield Mapmaking

First we focus on how the core configuration (see Figure 1) affects widefield mapmaking. We therefore plot the eigenvalue spectra of all three array configurations in Figure 6. In order to overcome the numerical difficult of evaluating ${{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}$ and its eigenspectrum, we pixelize the sky adaptively, using the hierarchical property of HEALPix to develop a pixelization scheme that roughly preserves equal primary beam power per pixel. This scheme ranges from N_side = 64 around zenith to N_side = 4 near the horizon. We perform this analysis for both a single snapshot and 40 minutes of rotation synthesis. We then calculate the eigensystem, normalizing by the number of antennas to make the comparison fair.

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At first glance, the eigenvalues—which encode the amount of information in each mode—are very similar, especially after Earth rotation synthesis. The biggest differences are revealed only in the case of instantaneous imaging, where we see a sharp cutoff for configuration (a) and a factor of a few difference between (b) and (c) in some of the more poorly measured eigenmodes.⁹ We expect that the performance of configuration (c) is due to the more even distribution of on-grid and off-grid baselines (see Figure 2).

To verify our intuition about the meaning of these eigenvalue spectra, we should look at the eigenvectors of the single snapshot. It is difficult to interpret or compare individual eigenvectors. Instead, we want to look at each ${{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}$ matrix in two different subspaces—one corresponding to the first 630 eigenvalues where, in Figure 6, the spectra are very similar and the other corresponding to the rest where they differ the most. In Figure 7 we show the diagonals of ${{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}$ for each array configuration, projected into the two orthogonal subspaces. One can think of these matrices as ${{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}$ where either the first 630 or the rest of the eigenvalues are set to zero.

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Roughly speaking, Figure 7 confirms our intuition that the highest information modes are those concentrated in the main lobe of the primary beam. That makes sense, since that is where the telescope is most sensitive. However, when we compare configuration (c) to either (a) or (b), we see a much clearer separation of information. The modes that contain information about the widefield are not the same modes as those that contain information about the primary field of view. It is not that configuration (c) is more sensitive to the modes deep in the primary beam. In fact, when when we add these two projected matrices back together, the results for all three arrays are nearly identical. Rather, what this result tells us is that we can make maps that more easily separate sources in the sidelobes from signal in the main lobe of the primary beam. As Thyagarajan et al. (2015c) and Pober et al. (2016) argue, these are the sources most likely to leak into the EoR window. The exact level of that isolation depends both on particular field of view and the observing frequency, both of which set the relative contributions of foregrounds in the main lobe and in the sidelobes of the primary beam.

3.3.2. High-resolution Mapmaking

We perform this analysis again on the full arrays (with outriggers) and obtain very similar results. In Figure 8 we assess the high-resolution mapmaking capability of each configuration by examining ${{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}$ for a 10° facet around at the zenith at the midpoint of the observation at high resolution (N_side = 256).

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We find that configurations (a) and (b), with their more compact cores, have more sensitivity to the best measured modes. By contrast, configuration (c) spreads its sensitivity out over a much larger number of independent modes. At very high eigenvalue number, we actually see configuration (a) performing well, due to its greater number of independently measured long baselines, although these modes are many orders of magnitude less sensitive than the best measured modes. Interestingly, even when rotation synthesis is included, the differences between the arrays are more persistent than they were in the widefield case. This is likely because the arrays differ more in their uv-coverage at long baselines than at short baselines, where the main hexagonal grid dominates.

Once again, by splitting the eigenspace of ${{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}$ for all three arrays at the eigenvalue crossing point of N_λ ≈ 1000, we can obtain a sense of where the information comes from. Just as in Figure 7, we again see in Figure 9 that configuration (c) has a better separation of information between the center and the edge of the main lobe of the primary beam. Configuration (a), with almost no off-grid baselines, cannot separate information between zenith and the widefield very well at all.

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Although the effects on mapmaking are modest, especially after Earth rotation synthesis, the separability of information between the main lobe of the primary beam and the sidelobes is interesting and potentially quite useful. In order to subtract foregrounds with sufficient fidelity to work within the wedge, we need precise subtraction of contaminants far from the main field of view (Thyagarajan et al. 2015c; Pober et al. 2016). The ability to instantaneously distinguish between foreground sidelobes and EoR signal is useful, especially when foreground positions and fluxes change due to the ionosphere (Yatawatta et al. 2013; Vedantham & Koopmans 2015, 2016; Ewall-Wice et al. 2016b). Likewise, the cleaner separation between modes will likely improve traditional sky model-based calibration or hybrid calibration schemes that rely on a sky model for bandpass calibration.

Fundamentally, the calibration problem for a radio interferometer can be expressed as¹⁰

$$\begin{eqnarray}&&{V}_{{ij}}^{\mathrm{measured}}={g}_{i}{g}_{j}^{* }{V}_{i-j}^{\mathrm{true}}+{n}_{{ij}}.\end{eqnarray} \tag{ 8 }$$

The visibility we measure, ${V}_{{ij}}^{\mathrm{measured}}$, is related to the true visibility by the product of the complex gain g_i of each antenna, plus some noise. We write the true visibility in shorthand as ${V}_{i-j}^{\mathrm{true}}$ to express the idea that all that matters is the baseline, the separation between antennas i and j, rather than the specific antennas involved.

The key idea behind redundant baseline calibration is that the number of numbers we want to estimate, the gains and true visibilities, is much smaller than the number of measurements we actually make. We have an overdetermined system. If we can linearize Equation (8), then we can write down a vector expression in the form of Equation (2) that relates all the visibilities measured to all the gains and true visibilities. Then we can use exactly the same statistical machinery to solve for both the gains and visibilities simultaneously.

Liu et al. (2010) propose two ways of doing that. The simpler way, called logcal, linearizes Equation (8) by taking the natural logarithm. If we express the complex gains as ${g}_{j}\equiv \exp [{\eta }_{j}+i{\phi }_{j}]$, then it follows that

$$\begin{eqnarray}&&\mathrm{ln}({V}_{{ij}}^{\mathrm{measured}})={\eta }_{i}+{\eta }_{j}+i{\phi }_{i}-i{\phi }_{j}+\mathrm{ln}({V}_{i-j}^{\mathrm{true}})+n{^{\prime} }_{{ij}}.\end{eqnarray} \tag{ 9 }$$

The precise form and statistics of n'_ij are discussed extensively in Liu et al. (2010). This can be rewritten in the same form as Equation (2) by treating ${\boldsymbol{y}}$ as the set of measurements, ${\boldsymbol{x}}$ as a vector that contains both the gains and the true visibilities, and ${\boldsymbol{A}}$ as the matrix whose rows specify which pairs of antennas with which gains measure which visibility. In this form the real and imaginary parts of the gains and visibilities, produce two different sets of equations.

It turns out that logcal's solutions are slightly biased. To rectify that, Liu et al. (2010) also developed lincal, a method of linearizing Equation (8) by Taylor expanding it and then solving for gains and visibilities iteratively. Both methods were implemented for Zheng et al. (2014) and were integrated into the PAPER analysis pipeline (Ali et al. 2015), with logcal providing lincal a good starting guess.¹¹

In our case, we are interested in how the array configuration affects the errors on redundant calibration. For simplicity, we look at the expected gain errors predicted by logcal. If we form an unbiased estimator of the gains and true visibilities, then the gain/visibility covariance, ${\boldsymbol{\Sigma }}$, is given by

$$\begin{eqnarray}&&{\boldsymbol{\Sigma }}={[{{\boldsymbol{A}}}^{\dagger }{{\boldsymbol{N}}}^{-1}{\boldsymbol{A}}]}^{-1}.\end{eqnarray} \tag{ 10 }$$

In the case where all baselines measure visibilities with the same magnitude¹² and all antennas have the same noise level, Liu et al. (2010) show that ${\boldsymbol{\Sigma }}$ is reduced to

$$\begin{eqnarray}&&{\boldsymbol{\Sigma }}\longrightarrow \displaystyle \frac{1}{{({\rm{S}}/{\rm{N}})}^{2}}{[{{\boldsymbol{A}}}^{\dagger }{\boldsymbol{A}}]}^{-1},\end{eqnarray} \tag{ 11 }$$

where S/N is the S/N of the sky signal (i.e., foregrounds), not just the 21 cm signal. This leads us to a new metric for assessing array configurations by looking at the gain errors predicted by the diagonal of ${[{{\boldsymbol{A}}}^{\dagger }{\boldsymbol{A}}]}^{-1}$. It can be shown that the Δηs we predict are simply the gain errors up to third order in Δη:

$$\begin{eqnarray}&&{\rm{\Delta }}| {g}_{i}| ={\rm{\Delta }}{\eta }_{i}+{ \mathcal O }({\rm{\Delta }}{\eta }^{3}).\end{eqnarray} \tag{ 12 }$$

There remains one important complication related to the essential nature of what redundant calibration can and cannot do. As it turns out, ${{\boldsymbol{A}}}^{\dagger }{\boldsymbol{A}}$ is actually never invertible. There must always be at least one zero-eigenvalue because there is always an overall amplitude degeneracy in this system. We can increase the gains by some factor, decrease the true visibilities by that factor squared, and still get the same measurements. For the imaginary half of logcal, it turns out that there are always at least three zero-eigenvalues, one overall phase and two tip/tilt terms related to the orientation of the array.¹³ This limitation is discussed extensively in Zheng et al. (2014).

This is not to say that redundant baseline calibration is not helpful. It reduces the problem of calibrating a gain and a phase for every antenna at every frequency to calibrating just a few numbers per frequency for the entire array. The latter problem is less noisy, less sensitive to the accuracy of the sky model, and therefore far less error-prone. Fortunately, the gain errors we examine are insensitive to the overall amplitude, since they are normalized by the S/N. We merely need to take the pseudoinverse of ${{\boldsymbol{A}}}^{\dagger }{\boldsymbol{A}}$ to obtain ${\boldsymbol{\Sigma }}$ (Dillon et al. 2013).

Before we look at the three array configurations considered above, let us try to develop some intuition for this formalism. In Figure 10 we plot the gain errors expected from redundant calibration of a 331-element solid hexagonal HERA core, the core of configuration (a). Plotted are the gain errors we would expect if each visibility had the same amplitude and were measured with a foreground S/N of 1. The errors depend inverse-linearly on the S/N.

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For a highly redundant configuration like the solid HERA core, the antenna-to-antenna variation in gain errors is very small. In these cases, the relative gain errors are all roughly just ${({N}_{\mathrm{antennas}})}^{-1/2}$ (Liu et al. 2010). However, we can see some illustrative structure in Figure 10. The edge and in particular corner antennas are are slightly more difficult to calibrate—they are involved in fewer highly redundant baselines than interior antennas. Likewise, antennas in the very center are involved in fewer kinds of baselines and thus also see slightly higher error. Both the redundancy of the baselines an antenna is involved in and the centrality of those baselines to the larger connected graph of redundant baselines affect the calibratability of each antenna. Regardless, for a 331-element core these variations are small in comparison to what we see for more complex array designs; the key comparison we wish to make is how well we can redundantly calibrate our full arrays relative to the a solid core.

After calculating the logcal ${{\boldsymbol{A}}}^{\dagger }{\boldsymbol{A}}$ for each of the three arrays we have considered above, configuration (a) sticks out right away. The imaginary version ${{\boldsymbol{A}}}^{\dagger }{\boldsymbol{A}}$ should have three zero-eigenvalue; it has five. Two extra phase factors are not redundantly calibratable because the outrigger baselines do not overlap as they do in configuration (b). While the core of configuration (a) can be redundantly calibrated, its outriggers cannot. We therefore cannot predict the errors from redundant calibration for this array.

Configurations (b) and (c), however, were both designed to be redundantly calibratable.¹⁴ They have the correct number of zero-eigenvalues, meaning that taking the pseudoinverse of ${{\boldsymbol{A}}}^{\dagger }{\boldsymbol{A}}$ is appropriate. In Figure 11 we show the expected errors on antenna gains when calibrating the cores redundantly and when calibrating the whole array redundantly.

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When calibrating just the cores, the addition of the 12 off-grid corner antennas in configuration (b) leads to higher gain errors not just in the off-grid antennas but also, surprisingly, in the hexagonal core by about 20%. By contrast, configuration (c) sees raised errors generally by less than 1% compared to a solid HERA core. This is because the core of configuration (b) probes many more baselines than a simple HERA core, but most of them are sparsely sampled. A large number of baselines are only weakly connected to the rest, leading to higher errors throughout. Of course, if one wishes to give up on redundant calibration of the off-grid antennas, one can achieve the same errors as in Figure 10. However, then one must resort to another strategy (i.e., one that requires a sky model) to calibrate the remaining antennas (which affect all the off-grid visibilities).

Those effects are mitigated somewhat by the addition of redundant outriggers that decrease the discrepancy between the array cores to a 5% difference in calibration errors. On the other hand, the most distant outriggers in configuration (b) have approximately 50% larger gain errors than the most furthest-flung outriggers in configuration (c). Simply put, the evenness of the distribution of redundancy across the baselines probed, and thus the connectedness of antennas through that network of redundant baselines, makes configuration (c) more redundantly calibratable.