Example: plot_ambig program

0 This program creates a scan of the ambiguity function.

To following is based on sections and . Using the definition () for the scalar product $\langle a,b \rangle$ we can rewrite the expectation value () of the signal to noise ratio (SNR) $\rho$ as

$$\langle \rho \rangle = 2 \frac{\vert\langle C, T_i \rangle_{t_0}\vert}{\sqrt{\vert T_i\vert}},$$ (7.18.166)

where $\vert T_i\vert = \sqrt{\langle T_i, T_i \rangle}$. Here ${C}(t)$ is the signal (i.e. the chirp), and $T_i$ is the i-th template. Obviously $\langle \rho \rangle_{t_0}$ is maximized if $T_i = C$ and $t_0=0$. We thus can rewrite equation () as

$$\langle \rho \rangle = \underbrace{\frac{\vert\langle C, ... ...rt }}}_{{\cal A}_{i t_0}} \, \langle \rho \rangle_{\rm max}.$$

The function ${\cal A}_{i t_0}$ gives the reduction of the SNR due to a nonoptimal template $T_i$. It is commonly called the ambiguity function. Since maximization over the parameter $t_0$ is trivially achieved by a FFT we often work with the reduced ambiguity function

$${\cal A}_i = \max_{t_0} {\cal A}_{i t_0}.$$

As was mentioned in section every signal is a linear combination of two orthogonal modes $T_0$ and $T_{90}$ (we suppress the index $i$ for now), where $\langle T_0, T_{90} \rangle = 0$. We can filter for any linear combination by using the template

$$T = \frac{1}{\sqrt{2}} \left( \frac{T_0}{\vert T_0\vert} + i \frac{T_{90}}{\vert T_{90}\vert} \right).$$

Using $T$, the ambiguity function becomes

$${\cal A}_i = \max_{t_0} \sqrt{ \frac{\langle C, T_0 \rangl... ...e C, T_{90} \rangle^2_{t_0}}{\vert T_{90}\vert \vert C\vert}}.$$ (7.18.167)

The sample program plot_ambig produces a file containing ${\cal A}_i$ as a function of the chirp mass ${\cal M} = (m_1 m_2)^{3/5} (m_1+m_2)^{-1/5}$ and the mass ratio $\eta = (m_1 m_2) (m_1+m_2)^{-2}$. The templates are taken to be the 2 pN spin-less wave forms and the signal $C$ is one of the modes calculated from perturbation theory. The output is saved to the file scan.dat. Includes/plot_ambig.tex

Figure: A contour plot of the reduced Ambiguity function ${\cal A}_i$. The axes are labeled by the relative deviations from the true values of $\eta=0.25$ and the chirp mass ${\cal M}=3.92 M_\odot$ corresponding to a $m_1=m_2=4.5 M_\odot$ binary system. The Maximum value of $86.8\%$ is attained at $\eta^* = 0.61 \eta$ and ${\it M}^* = 1.084 {\cal M}$.