LinearDimension¶

Generalized Class¶

Description¶

A LinearDimension is where the coordinates along the dimension follow a linear relationship with the indexes, \(\mathbf{J}_k\), along the dimension. Let \(\Delta x_k\) be the increment, \(N_k \ge 1\), the number of points (counts), \(b_k\), the coordinates offset, and \(o_k\), the origin offset along the \(k^{th}\) dimension, then the corresponding coordinates along the dimension, \(\mathbf{X}_k\), are given as

()¶\[\mathbf{X}_k = \Delta x_k (\mathbf{J}_k - Z_k) + b_k \mathbf{1},\]

and the absolute coordinates as,

()¶\[\mathbf{X}_k^\mathrm{abs} = \mathbf{X}_k + o_k \mathbf{1}.\]

Here, \(\mathbf{1}\) is an array of ones, and \(\mathbf{J}_k\) is the array of indexes along the \(k^\mathrm{th}\) dimension given as

()¶\[\mathbf{J}_k = [0, 1, 2, 3, ..., N_k-1].\]

The term, \(Z_k\), is an integer with a value of \(Z_k=0\) or \(\frac{T_k}{2}\) when the value of complex_fft attribute of the corresponding dimension object is false or true, respectively. Here, \(T_k=N_k\) and \(N_k-1\) for even and odd value of \(N_k\), respectively.

Note

When the value of the complex_fft attribute is true, and \(N_k\) is even, the dependent variable value corresponding to the index \(\pm N_k/2\) is an alias.

Attributes¶

Name	Type	Description
count	Integer	A required number of points, \(N_k\), along the dimension.
increment	ScalarQuantity	A required increment, \(\Delta x_k\), along the dimension.
coordinates_offset	ScalarQuantity	An optional coordinate, \(b_k\), corresponding to the zero of the indexes array, \(\bf{J}_k\). The default value is a physical quantity with zero numerical value.
origin_offset	ScalarQuantity	An optional origin offset, \(o_k\), along the dimension. The default value is a physical quantity with zero numerical value.
quantity_name	String	An optional quantity name associated with the physical quantities describing the dimension.
period	ScalarQuantity	An optional period of the dimension. By default, the dimension is considered non-periodic.
complex_fft	Boolean	An optional boolean with default value as False. If true, the coordinates along the dimension are evaluated as the output of a complex fast Fourier transform (FFT) routine. See the description.
reciprocal	ReciprocalDimension	An optional object with attributes required to describe the reciprocal dimension.

Example¶

The following LinearDimension object,

{
    "type": "linear",
    "count": 10,
    "increment": "2 µA",
    "coordinates_offset": "0.1 µA"
}

will generate a dimension, where the coordinates \(\mathbf{X}_k\) are

[
    "0.1 µA",
    "2.1 µA",
    "4.1 µA",
    "6.1 µA",
    "8.1 µA",
    "10.1 µA",
    "12.1 µA",
    "14.1 µA",
    "16.1 µA",
    "18.1 µA"
]