arXiv subjects

Samir Choudhuri

Publications and source records attributed to Samir Choudhuri.

At least 55 records · Page 3Linked to original sources

arXiv2020

Demonstrating the Tapered Gridded Estimator (TGE) for the Cosmological HI 21-cm Power Spectrum using $150 \, {\rm MHz}$ GMRT observations

We apply the Tapered Gridded Estimator (TGE) for estimating the cosmological 21-cm power spectrum from $150 \, {\rm MHz}$ GMRT observations which corresponds to the neutral hydrogen (HI) at redshift $z = 8.28$. Here TGE is used to measure the Multi-frequency Angular Power Spectrum (MAPS) $C_{\ell}(Δν)$ first, from which we estimate the 21-cm power spectrum $P(k_{\perp},k_{\parallel})$. The data here are much too small for a detection, and the aim is to demonstrate the capabilities of the estimator. We find that the estimated power spectrum is consistent with the expected foreground and noise behaviour. This demonstrates that this estimator correctly estimates the noise bias and subtracts this out to yield an unbiased estimate of the power spectrum. More than $47\%$ of the frequency channels had to be discarded from the data owing to radio-frequency interference, however the estimated power spectrum does not show any artifacts due to missing channels. Finally, we show that it is possible to suppress the foreground contribution by tapering the sky response at large angular separations from the phase center. We combine the k modes within a rectangular region in the `EoR window' to obtain the spherically binned averaged dimensionless power spectra $Δ^{2}(k)$ along with the statistical error $σ$ associated with the measured $Δ^{2}(k)$. The lowest $k$-bin yields $Δ^{2}(k)=(61.47)^{2}\,{\rm K}^{2}$ at $k=1.59\,\textrm{Mpc}^{-1}$, with $σ=(27.40)^{2} \, {\rm K}^{2}$. We obtain a $2 \, σ$ upper limit of $(72.66)^{2}\,\textrm{K}^{2}$ on the mean squared HI 21-cm brightness temperature fluctuations at $k=1.59\,\textrm{Mpc}^{-1}$.

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arXiv2020

First multi-redshift limits on post-Epoch of Reionization (post-EoR) 21 cm signal from z = 1.96 - 3.58 using uGMRT

Measurement of fluctuations in diffuse HI 21 cm background radiation from the post-reionization epoch (z < 6) is a promising avenue to probe the large-scale structure of the Universe and understand the evolution of galaxies. We observe the European Large-Area ISO Survey-North 1 (ELAIS-N1) field at 300-500 MHz using the upgraded Giant Meterwave Radio Telescope (uGMRT) and employ the 'foreground avoidance' technique to estimate the HI 21 cm power spectrum in the redshift range z = 1.96-3.58. Given the possible systematics that may remain in the data, we find the most stringent upper limits on the spherically averaged 21 cm power spectra at k~1.0 Mpc$^{-1}$ are (58.87 mK)$^2$, (61.49 mK)$^2$, (60.89 mK)$^2$, (105.85 mK)$^2$ at z = 1.96,2.19,2.62 and 3.58, respectively. We use this to constrain the product of neutral HI mass density (Omega_HI) and HI bias (b_HI) to the underlying dark matter density field, [Omega_HI*b_HI], as 0.09,0.11,0.12,0.24 at z=1.96,2.19,2.62,3.58, respectively. To the best of our knowledge these are the first limits on the HI 21 cm power spectra at the redshift range z = 1.96 - 3.58 and would play a significant role to constrain the models of galaxy formation and evolution.

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arXiv2020

Angular power spectrum of supernova remnants: effects of structure, geometry and diffuse foreground

The study of the intensity fluctuation power spectrum of individual supernova remnants (SNRs) can reveal the structures present at sub-pc scales, and also constrain the physical process that generates those structures. There are various effects, such as the remnant shell thickness, projection of a three-dimensional structure onto a two-dimensional observational plane, and the presence of diffuse "foreground" emission, which causes the observed power spectrum to deviate from the intrinsic power spectrum of the fluctuations. Here, we report results from a systematic study of these effects, using direct numerical simulations, in the measured power spectrum. For an input power-law power spectrum, independent of the power-law index, we see a break in the observed power law at a scale which depends on the shell thickness of a shell-type SNR, and the three-dimensional turbulence changes to two-dimensional turbulence beyond that scale. We also report how the estimated power spectrum is expected to deviate from the intrinsic SNR power spectrum in the presence of additional diffuse Galactic synchrotron emission (DGSE) around the remnant shell. For a reasonable choice of the parameters, if the intrinsic SNR power spectrum is shallower than the DGSE power spectrum, the SNR contribution dominates at small angular scales of the estimated power spectra. On the other hand, if the SNR power spectrum is relatively steeper, the original power spectra is recovered only over a small window of angular scales. This study shows how detailed modeling may be used to infer the true power spectrum from the observed SNR intensity fluctuations power spectrum, which in turn can be used to constrain the nature of the turbulence that gives rise to these small scale structures.

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arXiv2020

Characterizing EoR foregrounds: A study of the Lockman Hole Region at 325 MHz

One of the key science goals for the most sensitive telescopes, both current and upcoming, is the detection of the redshifted 21-cm signal from the Cosmic Dawn and Epoch of Reionization. The success of detection relies on accurate foreground modeling for their removal from data sets. This paper presents the characterization of astrophysical sources in the Lockman Hole region. Using 325 MHz data obtained from the GMRT, a $6^\circ \times 6^\circ$ mosaiced map is produced with an RMS reaching 50 $μ$Jy $\mathrm{beam}^{-1}$. A source catalog containing 6186 sources is created, and the Euclidean normalized differential source counts have been derived from it, consistent with previous observations as well as simulations. A detailed comparison of the source catalog is also made with previous findings - at both lower and higher frequencies. The angular power spectrum (APS) of the diffuse Galactic synchrotron emission is determined for three different galactic latitudes using the Tapered Gridded Estimator. The values of the APS lie between $\sim$1 mK$^2$ to $\sim$100 mK$^2$. Fitting a power law of the form $A\ell^{-β}$ gives values of $A$ and $β$ varying across the latitudes considered. This paper demonstrates, for the first time, the variation of the power-law index for diffuse emission at very high galactic locations. It follows the same trend that is seen at locations near the galactic plane, thus emphasizing the need for low-frequency observations for developing better models of the diffuse emission.

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arXiv2020

All sky angular power spectrum: I. Estimating brightness temperature fluctuations using TGSS 150 MHz survey

Measurements of the Galactic synchrotron emission is relevant for the 21-cm studies from the Epoch of Reionization. The study of the synchrotron emission is also useful to quantify the fluctuations in the magnetic field and the cosmic ray electron density of the turbulent interstellar medium (ISM) of our Galaxy. Here, we present the all-sky angular power spectrum $(C_{\ell})$ measurements of the diffuse synchrotron emission using the TIFR GMRT Sky Survey (TGSS) at 150 {\rm MHz}. We estimate $C_{\ell}$ using visibility data both before and after subtracting the modelled point sources. The amplitude of the measured $C_{\ell}$ falls significantly after subtracting the point sources, and it is also slightly higher in the Galactic plane for the residual data. The residual $C_{\ell}$ is most likely to be dominated by the Galactic synchrotron emission. The amplitude of the residual $C_{\ell}$ falls significantly away from the Galactic plane. We find the measurements are quite symmetric in the Northern and Southern hemispheres except in the latitude range $15-30^{\circ}$ which is the transition region from the disk dominated to diffuse halo dominated region. The comparison between this interferometric measurement with the scaled version of the Haslam rms map at 150 {\rm MHz} shows that the correlation coefficient $(r)$ is more than 0.5 for most of the latitude ranges considered here. This signifies the TGSS survey is quite sensitive to the diffuse Galactic synchrotron radiation.

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arXiv2019

Detailed study of ELAIS N1 field with the uGMRT -- II. Source Properties and Spectral Variation Of Foreground Power Spectrum from 300-500 MHz Observations

Understanding the low-frequency radio sky in depth is necessary to subtract foregrounds in order to detect the redshifted 21 cm signal of neutral hydrogen from the Cosmic Dawn, Epoch of Reionization (EoR) and post-reionization era. In this second paper of the series, we present the upgraded Giant Metrewave Radio Telescope (uGMRT) observation of the ELAIS N1 field made at 300-500 MHz. The image covers an area of $\sim 1.8$ $\mathrm{deg}^{2}$ and has a central background rms noise of $\sim$ 15 $μ\mathrm{Jy}$ $\mathrm{beam}^{-1}$. We present a radio source catalog containing 2528 sources (with flux densities > 100 $μ$Jy) and normalized source counts derived from that. The detailed comparison of detected sources with previous radio observations is shown. We discuss flux scale accuracy, positional offsets, spectral index distribution and correction factors in source counts. The normalized source counts are in agreement with previous observations of the same field, as well as model source counts from the Square Kilometre Array Design Study (SKADS) simulation. It shows a flattening below $\sim$1 mJy which corresponds to rise in population of star forming galaxies and radio-quiet AGN. For the first time, we estimated the spectral characteristics of the angular power spectrum or Multi-Frequency Angular Power Spectrum (MFAPS) of diffuse Galactic synchrotron emission (DGSE) over the wide frequency bandwidth of $300-500$~MHz from radio interferometric observations. This work demonstrates the improved capabilities of the uGMRT.

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arXiv2019

A study of Kepler supernova remnant: angular power spectrum estimation from radio frequency data

Supernova remnants (SNRs) have a variety of overall morphology as well as rich structures over a wide range of scales. Quantitative study of these structures can potentially reveal fluctuations of density and magnetic field originating from the interaction with ambient medium and turbulence in the expanding ejecta. We have used $1.5$GHz (L band) and $5$GHz (C band) VLA data to estimate the angular power spectrum $C_{\ell}$ of the synchrotron emission fluctuations of the Kepler SNR. This is done using the novel, visibility based, Tapered Gridded Estimator of $C_{\ell}$. We have found that, for $\ell = (1.9 - 6.9) \times 10^{4}$, the power spectrum is a broken power law with a break at $\ell = 3.3 \times 10^{4}$, and power law index of $-2.84\pm 0.07$ and $-4.39\pm 0.04$ before and after the break respectively. The slope $-2.84$ is consistent with 2D Kolmogorov turbulence and earlier measurements for the Tycho SNR. We interpret the break to be related to the shell thickness of the SNR ($0.35 $ pc) which approximately matches $\ell = 3.3 \times 10^{4}$ (i.e., $0.48$ pc). However, for $\ell > 6.9 \times 10^{4}$, the estimated $C_{\ell}$ of L band is likely to have dominant contribution from the foregrounds while for C band the power law slope $-3.07\pm 0.02$ is roughly consistent with $3$D Kolmogorov turbulence like that observed at large $\ell$ for Cas A and Crab SNRs.

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arXiv2019

Detailed study of the ELAIS N1 field with the uGMRT - I. Characterizing the 325 MHz foreground for redshifted 21 cm observations

In this first paper of the series, we present initial results of newly upgraded Giant Meterwave Radio Telescope (uGMRT) observation of European Large-Area ISO Survey-North 1 (ELAIS-N1) at 325 MHz with 32 MHz bandwidth. Precise measurement of fluctuations in Galactic and extragalactic foreground emission as a function of frequency as well as angular scale is necessary for detecting redshifted 21-cm signal of neutral hydrogen from Cosmic Dawn, Epoch of Reionization (EoR) and post-reionization epoch. Here, for the first time we have statistically quantified the Galactic and extragalactic foreground sources in the ELAIS-N1 field in the form of angular power spectrum using the newly developed Tapered Gridded Estimator (TGE). We have calibrated the data with and without direction-dependent calibration techniques. We have demonstrated the effectiveness of TGE against the direction dependent effects by using higher tapering of field of view (FoV). We have found that diffuse Galactic synchrotron emission (DGSE) dominates the sky, after point source subtraction, across the angular multipole range $ 1115 \leqslant \mathcal{\ell} \leqslant 5083 $ and $ 1565 \leqslant \mathcal{\ell} \leqslant 4754 $ for direction-dependent and -independent calibrated visibilities respectively. The statistical fluctuations in DGSE has been quantified as a power law of the form $\mathcal{C}_{\mathcal{\ell}}= A \mathcal{\ell}^{-β} $. The best fitted values of (A, $β$) are ($ 62 \pm 6$ $mK^{2}$, $2.55 \pm 0.3 $) and ($ 48 \pm 4$ $mK^{2}$, $2.28 \pm 0.4 $ ) for the two different calibration approaches. For both the cases, the power law index is consistent with the previous measurements of DGSE in other parts of sky.

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arXiv2018

A Tapered Gridded Estimator (TGE) for the Multi-Frequency Angular Power Spectrum (MAPS) and the Cosmological HI 21-cm Power Spectrum

In this work we present a new approach to estimate the power spectrum $P({\bf k})$ of redshifted HI 21-cm brightness temperature fluctuations. The MAPS $C_{\ell}(ν_a,ν_b)$ completely quantifies the second order statistics of the sky signal under the assumption that the signal is statistically homogeneous and isotropic on the sky. Here we generalize an already existing visibility based estimator for $C_{\ell}$, namely TGE, to develop an estimator for $C_{\ell}(ν_a,ν_b)$. The 21-cm power spectrum is the Fourier transform of $C_{\ell}(Δν)$ with respect to $Δν=\mid ν_a-ν_b \mid$, and we use this to estimate $P({\bf k})$. Using simulations of $150 \, {\rm MHz}$ GMRT observations, we find that this estimator is able to recover $P(k)$ with an accuracy of $5-20 \%$ over a reasonably large $k$ range even when the data in $80 \%$ randomly chosen frequency channels is flagged.

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arXiv2018

An Image-based Tapered Gridded Estimator (ITGE) for the angular power spectrum

We present the Image-based Tapered Gridded Estimator (ITGE) to measure the angular power spectrum $(C_{\ell})$ of the sky signal directly from the visibilities measured in radio-interferometric observations. The ITGE allows us to modulate the sky response through a window function which is implemented in the image plane, and it is possible to choose a wide variety of window functions. In the context of the cosmological HI 21-cm signal, this is useful for masking out the sky signal from specific directions which have strong residual foregrounds. In the context of the ISM in external galaxies, this is useful to separately estimate the $C_{\ell}$ of different parts of the galaxy. The ITGE deals with gridded data, hence it is computationally efficient. It also calculates the noise bias internally and exactly subtracts this out to give an unbiased estimate of $C_{\ell}$. We validate the ITGE using realistic VLA simulations at $1.4 {\rm GHz}$. We have applied the ITGE to estimate the $C_{\ell}$ of HI 21-cm emission from different regions of the galaxy NGC 628. We find that the slope of the measured $C_{\ell}$ in the outer region is significantly different as compared with the inner region. This indicates that the statistical properties of ISM turbulence possibly differ in different regions of the galaxy.

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arXiv2018

Turbulent power spectrum in warm and cold neutral medium using the Galactic HI 21 cm emission

Small-scale fluctuations of different tracers of the interstellar the medium can be used to study the nature of turbulence in astrophysical scales. Of these, the `continuum' emission traces the fluctuations integrated along the line of sight whereas, the spectral line tracers give the information along different velocity channels as well. Recently, Miville-Deschênes et al. (2016) have measured the intensity fluctuation power spectrum of the continuum dust emission, and found a power law behaviour with a power law index of $-2.9 \pm 0.1$ for a region of our Galaxy. Here, we study the same region using high-velocity resolution 21-cm emission from the diffuse neutral medium, and estimate the power spectrum at different spectral channels. The measured 21-cm power spectrum also follows a power law, however, we see a significant variation in the power law index with velocity. The value of the power-law index estimated from the integrated map for different components are quite different which is indicative of the different nature of turbulence depending on temperature, density and ionization fraction. We also measure the power spectra after smoothing the 21 cm emission to velocity resolution ranging from $1.03$ to $13.39~{\rm km~s^{-1}}$, but the power spectrum remains unchanged within the error bar. This indicates that the observed fluctuations are dominantly due to density fluctuations, and we can only constrain the power-law index of velocity structure function of $0.0 \pm 1.1$ which is consistent with the predicted Kolmogorov turbulence $(γ=2/3)$ and also with a shock-dominated medium $(γ=1.0)$.

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arXiv2017

Visibility-based Power Spectrum Estimation for Low-Frequency Radio Interferometric Observations

We present a visibility based estimator namely, the Tapered Gridded Estimator (TGE) to estimate the power spectrum of the diffuse sky signal. The TGE has three novel features. First, the estimator uses gridded visibilities to estimate the power spectrum which is computationally much faster than individually correlating the visibilities. Second, a positive noise bias is removed by subtracting the auto-correlation of the visibilities which is responsible for the noise bias. Third, the estimator allows us to taper the field of view so as to suppress the contribution from the sources in the outer regions and the sidelobes of the telescope's primary beam. We first consider the two dimensional (2D) TGE to estimate the angular power spectrum $C_{\ell}$. We have also extended the TGE to estimate the three dimensional (3D) power spectrum $P({\bf k})$ of the cosmological 21-cm signal. Analytic formulas are presented for predicting the variance of the binned power spectrum. Both the estimators and their variance predictions are validated using simulations of $150 \, {\rm MHz}$ GMRT observations. We have applied the 2D TGE to estimate $C_{\ell}$ using visibility data for two of the fields observed by TIFR GMRT Sky Survey (TGSS). We find that the sky signal, after subtracting the point sources, is likely dominated by the diffuse Galactic synchrotron radiation across the angular multipole range $240 \le \ell \lesssim 500$.

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arXiv2017

Validating a novel angular power spectrum estimator using simulated low frequency radio-interferometric data

The "Tapered Gridded Estimator" (TGE) is a novel way to directly estimate the angular power spectrum from radio-interferometric visibility data that reduces the computation by efficiently gridding the data, consistently removes the noise bias, and suppresses the foreground contamination to a large extent by tapering the primary beam response through an appropriate convolution in the visibility domain. Here we demonstrate the effectiveness of TGE in recovering the diffuse emission power spectrum through numerical simulations. We present details of the simulation used to generate low frequency visibility data for sky model with extragalactic compact radio sources and diffuse Galactic synchrotron emission. We then use different imaging strategies to identify the most effective option of point source subtraction and to study the underlying diffuse emission. Finally, we apply TGE to the residual data to measure the angular power spectrum, and assess the impact of incomplete point source subtraction in recovering the input power spectrum $C_{\ell}$ of the synchrotron emission. This estimator is found to successfully recovers the $C_{\ell}$ of input model from the residual visibility data. These results are relevant for measuring the diffuse emission like the Galactic synchrotron emission. It is also an important step towards characterizing and removing both diffuse and compact foreground emission in order to detect the redshifted $21\, {\rm cm}$ signal from the Epoch of Reionization.

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arXiv2017

The angular power spectrum measurement of the Galactic synchrotron emission in two fields of the TGSS survey

Characterizing the diffuse Galactic synchrotron emission at arcminute angular scales is needed to reliably remove foregrounds in cosmological 21-cm measurements. The study of this emission is also interesting in its own right. Here, we quantify the fluctuations of the diffuse Galactic synchrotron emission using visibility data for two of the fields observed by the TIFR GMRT Sky Survey (TGSS). We have used the 2D Tapered Gridded Estimator (TGE) to estimate the angular power spectrum $(C_{\ell})$ from the visibilities. We find that the sky signal, after subtracting the point sources, is likely dominated by the diffuse Galactic synchrotron radiation across the angular multipole range $240 \le \ell \lesssim 500$. We present a power law fit, $C_{\ell}=A\times\big(\frac{1000}{l}\big)^β$, to the measured $C_{\ell}$ over this $\ell$ range. We find that $(A,β)$ have values $(356\pm109~{\rm mK^2},2.8\pm0.3)$ and $(54\pm26~{\rm mK^2},2.2\pm0.4)$ in the two fields. For the second field, however, there is indication of a significant residual point source contribution, and for this field we interpret the measured $C_{\ell}$ as an upper limit for the diffuse Galactic synchrotron emission. While in both fields the slopes are consistent with earlier measurements, the second field appears to have an amplitude which is considerably smaller compared to similar measurements in other parts of the sky.

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arXiv2016

The prospects of measuring the angular power spectrum of the diffuse Galactic synchrotron emission with SKA1 Low

The diffuse Galactic synchrotron emission (DGSE) is the most important diffuse foreground component for future cosmological 21-cm observations. The DGSE is also an important probe of the cosmic ray electron and magnetic field distributions in the turbulent interstellar medium (ISM) of our Galaxy. In this paper we briefly review the Tapered Gridded Estimator (TGE) which can be used to quantify the angular power spectrum of the sky signal directly from the visibilities measured in radio-interferometric observations. The salient features of the TGE are (1.) it deals with the gridded data which makes it computationally very fast (2.) it avoids a positive noise bias which normally arises from the system noise inherent to the visibility data, and (3.) it allows us to taper the sky response and thereby suppresses the contribution from unsubtracted point sources in the outer parts and the sidelobes of the antenna beam pattern. We also summarize earlier work where the TGE was used to measure the C_l of the DGSE using 150 MHz GMRT data. Earlier measurements of the angular power spectrum are restricted to smaller angular multipole l ~ 10^3 for the DGSE, the signal at the larger l values is dominated by the residual point sources after source subtraction. The higher sensitivity of the upcoming SKA1 Low will allow the point sources to be subtracted to a fainter level than possible with existing telescopes. We predict that it will be possible to measure the angular power spectrum of the DGSE to larger values of l with SKA1 Low. Our results show that it should be possible to achieve l_{max} ~ 10^4 and ~ 10^5 with 2 minutes and 10 hours of observations respectively.

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arXiv2016

Imaging the redshifted 21-cm pattern around the first sources during the cosmic dawn using the SKA

Understanding properties of the first sources in the Universe using the redshifted \HI ~21-cm signal is one of the major aims of present and upcoming low-frequency experiments. We investigate the possibility of imaging the redshifted 21-cm pattern around the first sources during the cosmic dawn using the SKA1-low. We model the \HI ~21-cm image maps, appropriate for the SKA1-low, around the first sources consisting of stars and X-ray sources within galaxies. In addition to the system noise, we account also for the astrophysical foregrounds by adding them to the signal maps. We find that after subtracting the foregrounds using a polynomial fit and suppressing the noise by smoothing the maps over $10^{'} - 30^{'}$ angular scale, the isolated sources at $z \sim 15$ are detectable with $\sim 4 - 9 \, σ$ confidence level in 2000 h of observation with the SKA1-low. Although the 21-cm profiles around the sources get altered because of the Gaussian smoothing, the images can still be used to extract some of the source properties. We account for overlaps in the patterns of the individual sources by generating realistic \HI ~21-cm maps of the cosmic dawn that are based on $N$-body simulations and a one-dimensional radiative transfer code. We find that these sources should be detectable in the SKA1-low images at $z = 15$ with an SNR of $\sim 14 (4)$ in 2000 (200) h of observations. One possible observational strategy thus could be to observe multiple fields for shorter observation times, identify fields with SNR $\gtrsim 3$ and observe these fields for much longer duration. Such observations are expected to be useful in constraining the parameters related to the first sources.

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arXiv2016

The visibility based Tapered Gridded Estimator (TGE) for the redshifted 21-cm power spectrum

We present the improved visibility based Tapered Gridded Estimator (TGE) for the power spectrum of the diffuse sky signal. The visibilities are gridded to reduce the computation, and tapered through a convolution to suppress the contribution from the outer regions of the telescope's field of view. The TGE also internally estimates the noise bias, and subtracts this out to give an unbiased estimate of the power spectrum. An earlier version of the 2D TGE for the angular power spectrum $C_{\ell}$ is improved and then extended to obtain the 3D TGE for the power spectrum $P({\bf k})$ of the 21-cm brightness temperature fluctuations. Analytic formulas are also presented for predicting the variance of the binned power spectrum. The estimator and its variance predictions are validated using simulations of $150 \, {\rm MHz}$ GMRT observations. We find that the estimator accurately recovers the input model for the 1D Spherical Power Spectrum $P(k)$ and the 2D Cylindrical Power Spectrum $P(k_\perp,k_\parallel)$, and the predicted variance is also in reasonably good agreement with the simulations.

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arXiv2016

Tapering the sky response for angular power spectrum estimation from low-frequency radio-interferometric data

It is important to correctly subtract point sources from radio-interferometric data in order to measure the power spectrum of diffuse radiation like the Galactic synchrotron or the Epoch of Reionization 21-cm signal. It is computationally very expensive and challenging to image a very large area and accurately subtract all the point sources from the image. The problem is particularly severe at the sidelobes and the outer parts of the main lobe where the antenna response is highly frequency dependent and the calibration also differs from that of the phase center. Here we show that it is possible to overcome this problem by tapering the sky response. Using simulated 150 MHz observations, we demonstrate that it is possible to suppress the contribution due to point sources from the outer parts by using the Tapered Gridded Estimator to measure the angular power spectrum C_l of the sky signal. We also show from the simulation that this method can self-consistently compute the noise bias and accurately subtract it to provide an unbiased estimation of C_l.

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