# Observation matrix

*November 30, 2020*

# Introduction

To this date, CMB-S4 time-domain simulations have been based on the TOAST framework for time-ordered data simulation and analysis. This avails to the pipelines to all simulation and data reduction modules available in the TOAST framework. A shortcoming of these simulations has been the absence of a BICEP/Keck-style observation matrix that transforms input signal maps into output maps with the appropriate mode loss from time domain filtering.

We have developed a new TOAST operator, OpFilterBin, that applies a stack of filters to the TOD and bins a map. OpFilterBin can also calculate and output an observation matrix consistent with the filter stack. In this post we test OpFilterBin on a 10-day reference design tool simulation to establish accuracy and performance of the new operator.

# Mathematical

The TOAST filter-and-bin map is computed as

where is the pointing matrix, is the diagonal white noise matrix, is the filtering matrix:

and is the template matrix. Each column of is a template that gets projected out from the TOD. For the time being it consists of Legendre polynomials representing the ground and atmosphere. BICEP/Keck-style deprojection templates will be added in the future.