WDM.code.time_delay_filters package

Submodules

WDM.code.time_delay_filters.filters module

WDM.code.time_delay_filters.filters.time_delay_X(wdm: WDM_transform, n: int, nprime: int, m: int, mprime: int, delta_t: float) → float

Compute the time-delay matrix element \(X_{nn';mm'}(\delta t)\),

(69)\[X_{nn';mm'}(\delta t) = \int\mathrm{d}t g_{nm}(t+\delta t)g^*_{n'm'}(t).\]

This will return zero unless \(m'=m\), or \(m'=m\pm 1\).

Parameters:

• wdm (WDM.WDM_transform) – An instance of the WDM_transform class. This defines the wavelet basis.

• n (int) – The time index \(n\).

• nprime (int) – The time index \(n'\).

• m (int) – The frequency index \(m\).

• mprime (int) – The frequency index \(m'\).

• delta_t (float) – The time delay \(\delta t\), in the time units of wdm.

Returns:

X – The time-delay matrix element \(X_{nn';mm'}(\delta t)\).

Return type:

float

WDM.code.time_delay_filters.filters.time_delay_filter_Tl(wdm: WDM_transform, ell: int, delta_t: float) → float

The time-delay filter for the case \(m'=m\) is defined as

(70)\[T_{\ell}(\delta t)=\int\mathrm{d}f\exp(2\pi i f(\ell\Delta T-\delta t)) |\tilde{\Phi}(f)|^2 .\]

Parameters:

• wdm (WDM.WDM_transform) – An instance of the WDM_transform class. This defines the wavelet basis.

• ell (int) – The time index difference \(\ell=n-n'\).

• delta_t (float) – The time delay \(\delta t\), in the time units of wdm.

Returns:

T_l – The time-delay filter \(T_{\ell}(\delta t)\).

Return type:

float

WDM.code.time_delay_filters.filters.time_delay_filter_Tprimel(wdm: WDM_transform, ell: int, delta_t: float) → complex

The time-delay filter for the case \(m'=m\pm 1\) is defined as

(71)\[T'_{\ell}(\delta t)=\int\mathrm{d}f\exp(2\pi i f(\ell\Delta T-\delta t)) \tilde{\Phi}\left(f-\frac{1}{2}\Delta F\right) \tilde{\Phi}\left(f+\frac{1}{2}\Delta F\right) .\]

Parameters:

• wdm (WDM.WDM_transform) – An instance of the WDM_transform class. This defines the wavelet basis.

• ell (int) – The time index difference \(\ell=n-n'\).

• delta_t (float) – The time delay \(\delta t\), in the time units of wdm.

Returns:

Tprime_l – The time-delay filter \(T'_{\ell}(\delta t)\).

Return type:

float