Wilkinson Microwave Anisotropy Probe

Product Name
Optimal Time-Domain Filters


The optimal time-domain filters are used in the creation of the first estimate of the WMAP maximum likelihood sky map solution: (MT N-1 d) product, also called the iteration zero (it0) map. Final maps are produced by multiplying the it0 maps by (MT N-1 M)-1 through an iterative process.

The N-1 term represents the inverse radiometer noise correlation matrix and is computed using the inverse noise correlation function. This correlation function is the optimal time-domain filter. Please refer to Jarosik, et.at., 2007 for details on how the filter is used to create the inverse radiometer noise correlation matrix.

There are nine FITS files containing the collection of filters, one file for each year of data, and with each file containing the filters for all ten differencing assemblies. Each differencing assembly filter is written to its own FITS binary table extension, so there are ten such extensions in each file. The first column represents the lag time measured in samples of that radiometer. There are two columns that supply the the filters for each channel of that differencing assembly; the names of these columns names contain the radiometer channel names (e.g., FILTER_K11 and FILTER_K12, FILTER_W31 and FILTER_W32).

Filter Creation

Daily autocorrelation functions of time-ordered data were calculated over lag time from 0 to 104 s for each radiometer, using uncalibrated time-ordered data from which a linear interpolation of the hourly baseline was subtracted and an estimate of the sky signal in counts was subtracted (Hinshaw et al 2003). Yearly mean autocorrelation functions C(Δt) were formed from 60 days of data for each year. A cubic of the form

C(Δt)/C(0) = a + b log(Δt) + c [log(Δt)]2 + d [log(Δt)]3

was fit to the yearly mean data over lag time Δt from 2 times the sampling interval to 1000 s. Linear fits in log(Δt) were used for the K band radiometers, and quadratic fits were used for Ka; we make available the autocorrelation fit parameters in the form of an ASCII table.

Optimal time domain filters were made using the measured C(Δt)/C(0) at lag 1, the fit from lag 2 to the lag time where the fit crosses zero (from 400 to 1000 s), and C(Δt)/C(0) equal to zero for greater times. For each radiometer and each year, the filter was formed as follows:

  1. Fourier transform the model autocorrelation function to form the power spectral density P(f).
  2. Compute the Fourier space filter w(f) = 1/P(f) and Fourier transform it back into the time domain to form w(t), where t covers +- the zero-crossing time of the autocorrelation function.
  3. Form a zero-mean filter by subtracting the mean of w(t), and then multiply the entire filter by a normalization factor such that the response at zero lag is unity.

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