Harmonic filtering of the mask-restricted system ZS−1ZT. Top panel: inverse CMB power spectrum 1/Cℓ (solid blue), on the diagonal of the S−1 matrix behaves as ℓ2 near the beginning, increasing in steepness to ℓ8 at ℓ = 6000. It crosses the diagonal of the inverse-noise term N−1 (dotted green) at around ℓ = 1600. Above this point the system becomes prior dominated and the pseudo-inverse preconditioner works well both inside and outside the mask. Since we do not need the dedicated mask solver to solve for high ℓ, we apply a filter rℓ as described in Sect. 4.4, resulting in a filtered prior (dashed orange). Bottom panel: S−1 operator in pixel domain. The plot displays (YS−1YT)ij as a function of the distance between grid points i and j. Both the unfiltered operator 1/Cℓ (solid blue) and the low-pass filtered (dashed orange) have a “Mexican hat” or Wavelet-like azimuthally symmetric shape. The low-pass filter ensures that the oscillations decay quickly, making the system easier to solve in pixel domain than the unfiltered version.
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