Amplitude folding routines
[Pulsar Packages]

Collaboration diagram for Amplitude folding routines:

Contains function LALFoldAmplitudes: folds amplitudes into phase bins. More...
Periodic sources of gravitational radiation will produce measured strains of the following form:

\[ c[i] = A(t_i,\vec{\lambda}) \sin[\Phi(t_i,\vec{\lambda})] + n(t_i) \]

In this equation $c[i]$ is the discrete time series output of the detector (perhaps after some data conditioning, such as being resampled, narrow banded, or with instrument line noise removed). The amplitude, $A(t_i,\vec{\lambda})$, is assumed roughly constant at the gravity wave source, but is modulated by variation in the detector's response due to the Earth's motion. The phase, $\Phi(t_i,\vec{\lambda})$, is modulated by both the intrinsic spin down of the source, and the changes in relative motion between the source and the detector. This can be calculated for known pulsars. The vector $\vec{\lambda}$ is a vector of parameters that describe the sky position, etc., of the source and location, etc., of the detector. Finally, $n(t_i)$ is the noise, which also includes any other signals that are not coherent with the phase $\Phi(t_i,\vec{\lambda})$.

The folded amplitude is given by

\[ c_{\rm F} [j] = \sum_{i'} \left \{ A(t_i,\vec{\lambda})\sin[\Phi(t_i,\vec{\lambda})] + n(t_i) \right \} , \]

where the sum over $i'$ means sum over all $i$'s with $\Phi$ in phase bin $j$. If the bin sizes are sufficiently small, then $c_{\rm F} [j]$ can be approximated as

\[ c_{\rm F} [j] = \sin\Phi_j\sum_{i'} A(t_i,\vec{\lambda}) + \sum_{i'} n(t_i) , \]

where $\Phi_j$ is representive of the phase for bin $j$ (e.g., the phase corresponding to the midpoint of the bin). However, because of amplitude modulation, the amplitudes that are added to a phase bin are not guaranteed to enter with the same sign. Thus, some sort of amplitude demodulation should be done.

If we demodulate $A(t_i,\vec{\lambda})$ (for example, in a minimum way such as multiplying by the sign of the response function) we multiply each element of the vector $c[i]$ by an amplitude demodulation factor $D(t_i)$

\[ c_{\rm D\, , F} [j] = \sin\Phi_j \sum_{i'} D(t_i) A(t_i,\vec{\lambda}) + \sum_{i'} D(t_i) n(t_i) , \]

If the average value of $D(t_i)$ is zero, and is not correlated with the noise, then

\[ \sum_{i'} D(t_i) n(t_i) \approx 0 \]

However, the average value of $D(t_i)$ is probably not zero. The following is a very preliminary suggestion of how to further reduce the noise. Consider folding the measured strains, $c[i]$, again, but this time shifting the phase bins by $\pi$. Define this phase shifted folded amplitude as:

\[ c_{\pi, \, \rm D\, , F} [j] = \sin(\Phi_j + \pi) \sum_{i''} D(t_i) A(t_i,\vec{\lambda}) + \sum_{i''} D(t_i) n(t_i) , \]

where the sum over $i''$ means sum over all $i$'s with $\Phi + \pi$ in phase bin $j$. This will reverse the sign of the sum of the amplitudes that enter into each phase bin, but the sum of the noise contributions into each bin should be roughly the same. If the signal we are searching for is present, then amplitudes, $A(t_i,\vec{\lambda})$ are correlated with $D(t_i)$ such that

\[ \sum_{i'} D(t_i) A(t_i,\vec{\lambda}) \approx \bar{A} = {\rm constant} \]

Thus,

\[ c_{\rm D\, , F} [j] - c_{\pi, \, \rm D\, , F} [j] \approx 2 \bar{A}\sin\Phi_j , \]

plus residual noise. In practice, one needs to fold the amplitudes only once, and then make the replacement

\[ c_{\rm D\, , F} [j] \rightarrow c_{\rm D\, , F} [j] - c_{\rm D\, , F} [(j + N/2) \, \% \, j] , \]

where $N$ is the number of phase bins. We can then statistically analyze the hypothesis that the demodulated folded amplitudes correspond to a sinusoid.

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