Figure 1

Eigenvalues corresponding the different orders of DPSS tapers. Two cases with $NW=3$ and 6 are shown for $N=50$. Spectral concentration of the tapers rapidly worsen beyond $\alpha =2NW-1$.Reuse & Permissions

Figure 2

Examples of DPSS tapers and the corresponding spectral window functions for the case $N=50$ and $NW=6$. Upper panel: Real space form of the tapers of orders 0, 4, 6, and 8. Lower panel: The spectral window functions corresponding to the tapers in the upper panel. For simplicity, the window functions are shown only in the range $-3W<f<3W$ of frequency. The vertical dotted lines denote the edges of the bandwidth $(-W,W)$ within which the tapers are designed to be optimally concentrated. Tapers are ordered such that spectral leakage progressively increases for tapers of higher order.Reuse & Permissions

Figure 3

Top panel: The first 4 two-dimensional tapers on a $N\times N$ grid with $N=50$ and $NW=5$. The order of the one-dimensional tapers corresponding to each taper is indicated on the top. Bottom panel: The logarithm of the spectral window functions $\Gamma (k^1,k^2)$, corresponding to the tapers on the upper panel. The color scale on the lower panel are standardized to be $({10}^{-30},1)$ times the maximum in each plot. The white dotted lines represent the location of $\pm 2\pi W$ wave numbers.Reuse & Permissions

Figure 4

One realization of the CMB map on a $192\times 192\mathrm{pixel}$ grid. The physical size is 8° on a side. Estimation of the power spectrum is done on the $4°\times 4°$ subarea indicated by the rectangle. The color scale represents temperature fluctuations in micro Kelvin.Reuse & Permissions

Figure 5

Comparison of the simple periodogram method (labeled PM), the eigenvalue weighted MTM and the AMTM methods for estimating the power spectrum of a CMB map. In each plot, the continuous line represents the theory power spectrum used as an input for the Monte Carlo simulations. The open circles represent the mean values in each $\ell$ bin, averaged over 5000 realizations, while the vertical lines represent the $2\sigma$ spread. The bin width for this figure was $\Delta \ell =180$. For the MTM and AMTM methods $N_{\mathrm{tap}}=3^2$ tapers with $N_{\mathrm{res}}=2$ were used (see text for details). Note that the power spectra for the multitaper methods appear smoothed because they are convolved with the window function of an effective taper. Standard decorrelation techniques, like the MASTER algorithm [11] can be employed to debias and deconvolve all the above power spectra, but in the first two cases, where the mode-coupled power spectra are biased, decorrelation leads to bigger uncertainties in the deconvolved power spectra (see Sec. 5).Reuse & Permissions

Figure 6

Comparison of the probability distributions of the estimated power spectrum at three points (pixels) in the two-dimensional Fourier ($\ell$) space. From left to right, these points are $({\ell }^1,{\ell }^2)=(630,630)$, (1980, 630), and (2880, 630), for which the modulus $\ell$ has been indicated in each figure. Upper panel: Power spectrum estimation with $N_{\mathrm{res}}=2.0$ and $N_{\mathrm{tap}}=3^2$. The open circles (black), the diamonds (red), and the triangles (green) indicate the probability distribution of the quantity $\ell (\ell +1)C_{\ell }/(2\pi )$ as estimated via the AMTM, MTM, and PM methods from 5000 Monte Carlo simulations, respectively. The respective approximate theoretical forms as discussed in Sec. 2e are over plotted as the continuous curve (black), the dotted curve (red), and the dashed curve (green) for each of the methods. The mean degree of freedom of the chi-square for each method is also indicated as ${\nu }_A$ for AMTM, ${\nu }_M$ for MTM, and ${\nu }_P$ for the periodogram, PM. Each curve is also accompanied by a vertical line of the same style (and color) representing the mean value obtained from the Monte Carlo simulations. In each figure, the continuous (black) vertical line corresponding to the mean of the AMTM method, is also the value closest to the true power spectrum. It is actually the unbiased value of the pseudopower spectrum (see Sec. 5). Lower panel: Same as above but with $N_{\mathrm{res}}=3.0$ and $Ntap=5^2$.Reuse & Permissions

Figure 7

Comparison of the probability distributions of the estimated power spectrum in bins. Each panel is for a different combination of the $N_{\mathrm{res}}$ and $N_{\mathrm{tap}}$ as indicated on the top of the middle figure. The open circles (black), the diamonds (red) and the triangles (green) indicate the probability distribution of the quantity $\ell (\ell +1)C_{\ell }/(2\pi )$ in each bin as estimated via the AMTM, MTM and PM methods from 5000 Monte Carlo simulations, respectively. The respective approximate theoretical forms as discussed in Sec. 2e [see Eq. (37)] are over plotted as the continuous curve (black), the dotted curve (red), and the dashed curve (green) for each of the methods. The mean degree of freedom of the chi square for each method is also indicated as ${\nu }_A$ for AMTM, ${\nu }_M$ for MTM, and ${\nu }_P$ for the periodogram, PM. The number of pixels in each bin is also indicated as $N_b$. Note the periodogram is absent in each of the right-most plots, as it is highly biased and lies outside the range plotted.Reuse & Permissions

Figure 8

Covariance matrix of the bandpowers estimated via AMTM for three different parameter settings; from left to right these are $N_{\mathrm{res}}=2.0$, $N_{\mathrm{tap}}=3$; $N_{\mathrm{res}}=3.0$, $N_{\mathrm{tap}}=5$ and $N_{\mathrm{res}}=4.0$, $N_{\mathrm{tap}}=4$. Each square in the image represents a bin in $\ell$. For the $N_{\mathrm{res}}=2.0$ and $N_{\mathrm{res}}=3.0$ cases, we chose the bin-widths to be twice the fundamental resolution element, i.e. $\Delta \ell =2{\ell }_{\mathrm{fund}}=180$, while for the $N_{\mathrm{res}}=4.0$ case it was taken to be $4{\ell }_{\mathrm{fund}}=360$. There is appreciable covariance only between bins inside the resolution $\Delta {\ell }_W=2N_{\mathrm{res}}{\ell }_{\mathrm{fund}}$ set by the taper, and the covariance drops drastically beyond that frequency.Reuse & Permissions

Figure 9

“Disc-difference” function $W_{R-3R}$ discussed in the text (solid line). The dashed line represents the function $x^4$.Reuse & Permissions

Figure 10

Effect of disc differencing and self-injection on the power spectrum. The dotted line is the true power spectrum with a large dynamic range. The disc-differencing operation alone produces the dashed curve, which has a much smaller dynamic range, but is steeply rising at low multipoles. The disc radius used is $R=1^{\prime }$. The dotted-dashed curve shows the true power spectrum multiplied by a constant ${\alpha }^2$ where $\alpha =0.02$. If we disc difference the map followed by self-injection of a fraction $\alpha$ of the map, then the power spectrum of the processed map is the solid curve [given by (51)], which is conveniently flat over the range of multipoles.Reuse & Permissions

Figure 11

Prewhitening and AMTM as a remedy to aliasing of power due to point source mask. The dotted curve represents the input power spectrum from which the map is generated. The triangles represent the recovered AMTM power spectrum ($N_{\mathrm{res}}=3.0$, $N_{\mathrm{tap}}=3^2$) of the map after a point source mask is applied directly to it. If, on the other hand, the map is first prewhitened (see text) and the mask is subsequently applied, one obtains the diamonds as the AMTM power spectrum. The solid curve is the theoretical prediction for the prewhitened power spectrum. Thereafter, one divides the spectrum by the prewhitening transfer function [see (51)], obtaining a nearly unbiased estimate, denoted by the open circles. Note that the AMTM power spectra appear smoother than the true spectra as the former is convolved with the window function of the taper.Reuse & Permissions

Figure 12

Same as in Fig. 11, but for a map with white noise. In addition to the standard prewhitening operation, a Gaussian smoothing has been applied to flatten the tail of the prewhitened map.Reuse & Permissions

Figure 13

Deconvolution of the power spectrum. Left panel: Deconvolution of the periodogram with top-hat weighting. The black squares represent the binned power spectrum obtained directly from the map using the periodogram (straight FT) method. As discussed in the text, this power spectrum is the true power spectrum convolved with the mode-mode coupling matrix due to the top hat, the theoretical expectation for which is displayed as the blue line. The red points represent the binned power spectrum deconvolved via (60). The black line is the input power spectrum. The deconvolved binned power spectrum is to be compared with the binned input power spectrum, which is displayed as the black histogram. All points displayed are the mean of 800 Monte Carlo simulations and the error bars correspond to the $2\sigma$ spread in their values. Right panel: Same as above, but for the AMTM method. The mode-coupled power spectrum and the corresponding theoretical curve in this case have been artificially shifted below the deconvolved power spectra for easy viewing. As discussed in the text, the mode-coupled power spectrum produced by the AMTM method is a nearly unbiased estimate of the true power spectrum, while the mode-coupled periodogram (left panel) is highly biased at large multipoles. This bias causes the error bars in the deconvolved periodogram to be much larger than those in the deconvolved AMTM power spectrum at large $\ell$, as shown in Fig. 14.Reuse & Permissions

Figure 14

Fractional errors in the deconvolved binned power spectrum. The open circles represent the fractional errors for the periodogram method (left panel of Fig. 13). The filled circles represent the same for the AMTM method (right panel of Fig. 13). Although the deconvolved power spectrum obtained from either method is an unbiased estimate of the true power spectrum, the errors from the periodogram method are much larger at high $\ell$ because of the highly biased nature of the mode-coupled periodogram at those multipoles.Reuse & Permissions

Figure 15

Effect of prewhitening on the errors in the deconvolved power spectrum in presence of a point source mask. Top panel: For the AMTM method: The open circles represent the fractional $1\sigma$ error bars on the deconvolved power spectrum when prewhitening is not performed, showing that holes in the point source mask render the AMTM method biased at high multipoles and lead to large error bars. The filled circles represent the same after prewhitening has been performed, showing that prewhitening remedies the leakage of power and makes the power spectrum estimator nearly unbiased. Bottom panel: The same as above, but for the periodogram or straight FT. Note that prewhitening, if performed properly, makes the periodogram as good a power spectrum tool as the AMTM.Reuse & Permissions