rss_ringoccs.diffrec.window_functions module¶

Purpose:

Provide a suite of window functions and functions related to the normalized equivalent width of a given array.

Dependencies:

numpy
spicy

rss_ringoccs.diffrec.window_functions.coss(x, W)¶

Purpose:

Compute the Cosine Squared Window Function.

Arguments:

x (np.ndarray):	Real valued array used for the independent variable, or x-axis.
w_in (float):	Width of the window. Positive float.

Outputs:

w_func (np.ndarray):
	Window function of width w_in evaluated along x.

rss_ringoccs.diffrec.window_functions.kb20(x, W)¶

Purpose:

Compute the Kaiser-Bessel 2.0 Window Function.

Arguments:

x (np.ndarray):	Real valued array used for the independent variable, or x-axis.
w_in (float):	Width of the window. Positive float.

Outputs:

w_func (np.ndarray):
	Window function of width w_in evaluated along x.

rss_ringoccs.diffrec.window_functions.kb25(x, W)¶

Purpose:

Compute the Kaiser-Bessel 2.5 Window Function.

Arguments:

x (np.ndarray):	Real valued array used for the independent variable, or x-axis.
w_in (float):	Width of the window. Positive float.

Outputs:

w_func (np.ndarray):
	Window function of width w_in evaluated along x.

rss_ringoccs.diffrec.window_functions.kb35(x, W)¶

Purpose:

Compute the Kaiser-Bessel 3.5 Window Function.

Arguments:

x (np.ndarray):	Real valued array used for the independent variable, or x-axis.
w_in (float):	Width of the window. Positive float.

Outputs:

w_func (np.ndarray):
	Window function of width w_in evaluated along x.

rss_ringoccs.diffrec.window_functions.kbal(x, W, alpha)¶

Purpose:

Create a Kaiser-Bessel window with a user specified alpha parameter.

Variables:

W (float):	Window width.
dx (float):	Width of one point.
al (float):	The alpha parameter $\alpha_0$ .

Outputs:

w_func (np.ndarray):
	The Kaiser-Bessel alpha window of width w_in and spacing dx between points.

Notes:

The Kaiser-Bessel window is computed using the modified Bessel Function of the First Kind. It’s value is $y = I_0(\alpha\sqrt{1-4x^2/w^2})/I_0(\alpha)$ , where w is the window width.
We automatically multiply the alpha parameter by pi, so the kbal window function has an alpha value of $\alpha = \alpha_0 \pi$
The endpoints of the Kaiser-Bessel function tend to zero faster than $(1+2\alpha) / \exp(\alpha)$

Warnings:

None of the Kaiser-Bessel windows evaluate to zero at their endpoints. The endpoints are $1/I_0(\alpha)$ . For small values of alpha this can create Gibb’s like effects in reconstruction do to the large discontinuity in the window. For alpha values beyond $2.5\pi$ this effect is negligible.

rss_ringoccs.diffrec.window_functions.kbmd20(x, W)¶

Purpose:

Compute the Modified Kaiser-Bessel 2.0 Window Function.

Arguments:

x (np.ndarray):	Real valued array used for the independent variable, or x-axis.
w_in (float):	Width of the window. Positive float.

Outputs:

w_func (np.ndarray):
	Window function of width w_in evaluated along x.

rss_ringoccs.diffrec.window_functions.kbmd25(x, W)¶

Purpose:

Compute the Modified Kaiser-Bessel 2.5 Window Function.

Arguments:

x (np.ndarray):	Real valued array used for the independent variable, or x-axis.
w_in (float):	Width of the window. Positive float.

Outputs:

w_func (np.ndarray):
	Window function of width w_in evaluated along x.

rss_ringoccs.diffrec.window_functions.kbmd35(x, W)¶

Purpose:

Compute the Modified Kaiser-Bessel 3.5 Window Function.

Arguments:

x (np.ndarray):	Real valued array used for the independent variable, or x-axis.
w_in (float):	Width of the window. Positive float.

Outputs:

w_func (np.ndarray):
	Window function of width w_in evaluated along x.

rss_ringoccs.diffrec.window_functions.kbmdal(x, W, al)¶

Purpose:

Create a modifed Kaiser-Bessel window with a user specified alpha parameter.

Variables:

W (float):	Window width.
dx (float):	Width of one point.
al (float):	The alpha parameter $\alpha_0$ .

Outputs:

w_func (np.ndarray):
	The Kaiser-Bessel alpha window of width and w_in spacing dx between points.

Notes:

The Kaiser-Bessel window is computed using the modified Bessel Function of the First Kind. It’s value is $y = I_0(\alpha\sqrt{1-4x^2/w^2})/I_0(\alpha)$ , where w is the window width.
We automatically multiply the alpha parameter by pi, so the kbal window function has an alpha value of $\alpha = \alpha_0\pi$
The endpoints of the Kaiser-Bessel function tend to zero faster than $(1+2\alpha)) / \exp(\alpha)$

Warnings:

None of the Kaiser-Bessel windows evaluate to zero at their endpoints. The endpoints are $1/I_0(\alpha)$ . For small values of alpha this can create Gibb’s like effects in reconstruction due to the large discontinuity in the window. For alpha values beyond $2.5\pi$ this effect is negligible.

rss_ringoccs.diffrec.window_functions.normalize(dx, ker, f_scale)¶

Purpose:

Compute the window normalization

Arguments:

ker (np.ndarray):
	The Fresnel Kernel.
dx (float):	The spacing between points in the window. This is equivalent to the sample spacing. This value is in kilometers.
f_scale (np.ndarray):
	The Fresnel Scale in kilometers.

Outputs:

norm_fact (float):
	The normalization of the input Fresnel Kernel.

rss_ringoccs.diffrec.window_functions.rect(x, W)¶

Purpose:

Compute the rectangular window function.

Arguments:

x (np.ndarray):	Real valued array used for the independent variable, or x-axis.
w_in (float):	Width of the window. Positive float.

Outputs:

w_func (np.ndarray):
	Window function of width w_in evaluated along x.

rss_ringoccs.diffrec.window_functions.window_width(res, normeq, fsky, fres, rho_dot, sigma, bfac=True, Return_P=False)¶

Purpose:

Compute the window width as a function of ring radius. This is given from MTR86 Equations 19, 32, and 33.

Variables:

fsky (float or np.ndarray):
res (float):	The requested resolution.
normeq (float):	The normalized equivalent width. Unitless.
	The sky frequency.
fres (float or np.ndarray):
	The Fresnel scale.
rdot (float) or (np.ndarray):
	The time derivative of the ring radius.

Output:

w_vals (np.ndarray):
	The window width as a function of ring radius.