TOI processing LFI

ADC Correction

More on P02 and P02a.

Application

For each of the 44 LFI diodes there is the corresponding object in the Database. Each object contains 4 columns: the input voltages coming from the sky channel and the corresponding linearized output, the input voltages coming from the reference channel and the corresponding linearized output.

Data loaded by the module are used to initialize two different interpolator using CSPLINE and the functions from gsl (GNU Scientific Libraries) libraries. The interpolators are then used to correct each sample.

Spikes Removal

Some of the LFI receivers exhibit a small artifact, visible in the power spectra. The effect is a set of spikes at 1 Hz and harmonics. More information can be found in P02 and P02a.

Modelization

The method to estimate the 1 Hz signal is to make use of linear filtering of a template. The first step is dividing the data into bins: this is done by taking fractional part of the sample time in seconds (using on-board time) and placing the time sample into a bin depending upon the particular fraction of a second in which it lays. The number of bins is computed using:

[math] nbins = fsamp * template\_resolution[/math]

where fsamp is the sampling frequency, while template resolution is 136 at 70 GHz, 80 at 44 GHz and 56 at 30 GHz. Then the bins vector is initialized with time intervals.

We can write the process adding an index to the time sample: lower index denotes the particular time sample, while the upper index labels the bin into which the sample falls. The linear filter can be written as:

[math] s(t_{i}^{j}) = a_j \left(1- \Delta x (t_{i}^{j}) \right) + a_{j+1} \Delta x (t_{i}^{j})[/math]

Here [math] \Delta x (t_{i}^{j})[/math] is the filter weight which is determined by where within the bin sample lies. If we use [math] t^j [/math] with only an upper index to denote the start of each bin, then we can write the filter wright as follows:

[math] \Delta x (t_{i}^{j}) = {{{t_i^j - t^j} \over {t^{j+1} - t^j}}} [/math]

In other words, the filter weight is the time sample value minus the start of the bin divided by the width of the bin.

We must estimate the parameters [math] a_j [/math] from the data. With the assumption that the instrument has stable noise properties, we can use a least square algorithm to estimate the bin values:

[math] {\partial \over \partial a_k} \sum_{i,j} \left( s(t_i^j) – d_i^j \right)^2 = 0 [/math]

This can be represented in matrix equation:

[math] M_{jk}a_k = b_j [/math]

with the following definitions:

[math] M_{k,k-1} = \sum_i (1 - \Delta x (t_i^{k-1})) \Delta x (t_i^{k-1}) [/math]

[math] M_{k,k} = \sum_i (1 - \Delta x (t_i^k))^2 \Delta x (t_i^{k-1})^2 [/math]

[math] M_{k,k+1} = \sum_i (1 - \Delta x (t_i^k)) \Delta x (t_i^k) [/math]

[math] M_{k,k+n} (|n| \gt 1) = 0 [/math]

[math] b_j = \sum_i d_i^k (1- \Delta x (t_k^i)) + d_i^{k-1}\Delta x (t_i^{k-1}) [/math]

With that definitions we have to make use of periodic boundary conditions to obtain the correct results, such that if [math] k = 0 [/math], [math] k-1 = n-1 [/math] and [math] k = n-1 [/math], [math] k+1 = 0 [/math]. Once this is done, we have a symmetric tridiagonal matrix with additional values at the upper right and lower left corners of the matrix. The matrix is solved with LU decomposition. In order to be certain of the numerical accuracy of the result, we can perform a simple iteration. The solving of the linear system and the iterative improvement of the solution are implemented as suggested in Numerical Recipes.

Application

For each of the 44 LFI diodes there is the corresponding object in the Database. Because of the amplitude of the spikes we choose to apply correction only on the 44 GHz radiometers. Each object contains 3 columns: the bins start time vector, the sky amplitudes and the reference amplitudes.

For each sample the value to be subtracted is computed using:

[math] V = skyAmp_k (1 - \Delta x (t_k)) + skyAmp_{k+1} \Delta x (t_k) [/math]

where k is the index of the bins at a given time.