Math

spinlab.math.lineshape.gaussian(x, x0, sigma, integral=1.0)

Gaussian distribution.

Parameters:

• x (array_like) -- input x

• x0 (float) -- Center of distribution

• sigma (float) -- Standard deviation of Gaussian distribution

• integral (float) -- Integral of distribution

Returns:

Gaussian distribution

Return type:

ndarray

The Gaussian distribution is defined as:

\[f(x; x_0, \sigma) = \frac{1}{\sigma \sqrt{2 \pi}} \exp{\left(\frac{(x-x_0)^2}{2 \sigma^2}\right)}\]

spinlab.math.lineshape.lorentzian(x, x0, gamma, integral=1.0, deriv=False)

Lorentzian Distribution.

Parameters:

• x (array_like) -- input x

• x0 (float) -- Center of distribution

• gamma (float) -- Lorentzian width. 2*gamma is full width at half maximum (FWHM)

• integral (float) -- Integral of distribution

• deriv (boolean) -- Derivative of a Lorentzian Distribution (Imaginary part of a phased spectrum)

Returns:

Lorentzian distribution

Return type:

ndarray

The Lorentzian distribution is defined as:

\[f(x) = \frac{1}{\pi \gamma} \left[\frac{\gamma^2}{(x-x_0)^2 + \gamma^2}\right]\]

Derivative:

\[f(x) = \frac{1}{\pi \gamma} \left[\frac{- 2\gamma^2 (x-x_0)}{\left( (x-x_0)^2 + \gamma^2 \right)^2}\right]\]

spinlab.math.lineshape.voigtian(x, x0, sigma, gamma, integral=1.0, deriv=False)

Voigtian distribution. Lineshape given by a convolution of Gaussian and Lorentzian distributions.

Parameters:

• x (array_like) -- input x

• x0 (float) -- center of distribution

• sigma (float) -- Gaussian Linewidth. Standard deviation of Gaussian distribution.

• gamma (float) -- Lorentzian linewidth. 2*gamma is the full width at half maximum (FWHM)

• integral (float) -- Integral of distribution

• deriv (boolean) -- Derivative of a Voigtian distribution (Gaussian broadened imaginary part of a phased spectrum).

Returns:

Voigtian distribution

Return type:

ndarray

The Voigtian distribution is defined as:

\[f(x; x_0, \sigma, \gamma) = \frac{\operatorname{Re}[w(z)]}{\sigma \sqrt{2 \pi}}\]

with

\[z = \frac{x + i\gamma}{\sigma \sqrt{2}}\]

Derivative: .. math:

f(x) = \frac{1}{\sigma^3 \sqrt{2 \pi}} \left[ \gamma \operatorname{Im}[w(z)] - \left(x - x0\right) \operatorname{Re}[w(z)] \right]

with

\[z = \frac{\left( \left( x - x0 \right) + 1j \gamma \right)}{\sigma \sqrt{2}}\]

spinlab.math.relaxation.buildup_function(p, E_max, p_half)

Calculate asymptotic buildup curve

Parameters:

• p (array) -- power series

• E_max (float) -- maximum enhancement

• p_half (float) -- power at half saturation

Returns:

buildup curve

Return type:

ndarray

\[f(p) = 1 + E_{max} * p / (p_{1/2} + p)\]

spinlab.math.relaxation.general_biexp(t, C1, C2, tau1, C3, tau2)

Calculate bi-exponential curve

Parameters:

• t (array_like) -- time series

• C1 (float) -- see equation

• C2 (float) -- see equation

• C3 (float) -- see equation

• tau1 (float) -- see equation

• tau2 (float) -- see equation

Returns:

bi-exponential curve

Return type:

ndarray

\[f(t) = C1 + C2 e^{-t/tau1} + C3 e^{-t/tau2}\]

spinlab.math.relaxation.general_exp(t, C1, C2, tau)

Calculate mono-exponential curve

Parameters:

• t (array_like) -- time series

• C1 (float) -- see equation

• C2 (float) -- see equation

• tau (float) -- see equation

Returns:

mono-exponential curve

Return type:

ndarray

\[f(t) = C1 + C2 e^{-t/tau}\]

spinlab.math.relaxation.ksigma_smax(p, E_max, p_half)

Calculate asymptotic buildup curve

Parameters:

• p (array) -- power series

• E_max (float) -- maximum enhancement

• p_half (float) -- power at half saturation

Returns:

buildup curve

Return type:

ndarray

\[f(p) = E_{max} * p / (p_{1/2} + p)\]

spinlab.math.relaxation.logistic(x, c, x0, L, k)

Not Implemented. Placeholder for calculating asymptotic buildup curve

Parameters:

• x (array) -- x values

• c (float) -- offset

• x0 (float) -- x-value of sigmoid's midpoint

• L (float) -- maximum value

• k (float) -- logistic growth steepness

Returns:

buildup curve

Return type:

ndarray

spinlab.math.relaxation.t1(t, T1, M_0, M_inf)

Exponential recovery for inversion recovery and saturation recovery T1 Measurements

Parameters:

• t (array_like) -- time series

• T_1 (float) -- T1 value

• M_0 (float) -- see equation

• M_inf (float) -- see equation

Returns:

T1 curve

Return type:

ndarray

\[f(t) = M_{\infty} - (M_{\infty} - M_0) e^{-t/T_1}\]

spinlab.math.relaxation.t2(t, M_0, T2, p=1.0)

Calculate stretched or un-stretched (p=1) exponential T2 curve

Parameters:

• t (array_like) -- time series

• M_0 (float) -- see equation

• T_2 (float) -- T2 value

• p (float) -- see equation

Returns:

T2 curve

Return type:

ndarray

\[f(t) = M_{0} e^{(-(t/T_{2})^{p}}\]

spinlab.math.window.exponential(x, lw)

Calculate exponential window function

Parameters:

• x (array_like) -- Vector of points

• lw (int or float) -- Exponential line broadening.

Returns:

exponential window function

Return type:

array

\[\mathrm{exponential}(x) = e^{-\pi (x - x_0) lw}\]

spinlab.math.window.gaussian(x, lw)

Calculate gaussian window function

Parameters:

• x (array_like) -- vector of points

• lw (float) -- Full width at half maximum of the Gaussian window.

Returns:

gaussian window function

Return type:

array

\[\begin{split}\sigma &= \frac{lw}{2\sqrt{2\ln(2)}} \\ \mathrm{gaussian}(x) &= e^{-2\pi^2 x^2 \sigma^2}\end{split}\]

spinlab.math.window.hamming(x)

Calculate hamming window function

Parameters:

x (array_like, int) -- vector of points or number of points.

Returns:

hamming window function

Return type:

ndarray

\[\mathrm{hamming} = 0.53836 + 0.46164\cos(\pi * n / (N-1))\]

spinlab.math.window.hann(x)

Calculate hann window function

Parameters:

x (array_like, int) -- vector of points or number of points.

Returns:

hann window function

Return type:

ndarray

\[\mathrm{hann} = 0.5 + 0.5\cos(\pi * n / (N-1))\]

spinlab.math.window.lorentz_gauss(x, lw, gauss_lw, gaussian_max=0)

Calculate lorentz-gauss window function

Parameters:

• x (array_like) -- vector of points

• lw (int or float) -- exponential linewidth

• gauss_lw (int or float) -- gaussian linewidth

• gaussian_max (int or float) -- location of maximum in gaussian window

Returns:

gauss_lorentz window function

Return type:

array

\[ \begin{align}\begin{aligned}\mathrm{lorentz\_gauss} &= \exp(L - G^{2}) &\\ L(t) &= \pi * \mathrm{linewidth[0]} * t &\\ G(t) &= 0.6\pi * \mathrm{linewidth[1]} * (\mathrm{gaussian\_max} * (N - 1) - t) &\end{aligned}\end{align} \]

spinlab.math.window.sin2(x)

Calculate sin-squared window function

Parameters:

x (array_like, int) -- vector of points or number of points.

Returns:

sin-squared window function

Return type:

array

\[\sin^{2} = \cos((-0.5\pi * n / (N - 1)) + \pi)^{2}\]

spinlab.math.window.traf(x, lw)

Calculate traf window function

Parameters:

• x (array_like) -- vector of points

• lw (int or float) -- linewidth of Traficante window

Returns:

traf window function

Return type:

ndarray

\[ \begin{align}\begin{aligned}\mathrm{traf} &= (f1 * (f1 + f2)) / (f1^{2} + f2^{2}) &\\ f1(t) &= \exp(-t * \pi * \mathrm{linewidth[0]}) &\\ f2(t) &= \exp((t - T) * \pi * \mathrm{linewidth[1]}) &\end{aligned}\end{align} \]

The shaped pulses simulation

Author: Timothy Keller

Edit: Yen-Chun Huang

spinlab.math.pulses.adiabatic(tp, BW, beta, resolution=1e-09)

Make Adiabatic Pulse Shape based on Hyperbolic Secant pulse

\[\text{sech} \left( \beta (t - \frac{t_p}{2}) \right) ^{1+i(\pi BW / \beta)}\]

Parameters:

• tp (float) -- pulse length

• BW (float) -- pulse bandwidth

• beta (float) -- truncation factor

• resolution (float) -- pulse resolution

Returns:

tuple containing:

t (numpy.ndarray): Time axes

pulse (numpy.ndarray): Pulse shape

Return type:

tuple

spinlab.math.pulses.chirp(tp, BW, resolution=1e-09)

Complex chirp pulse

\[e^{i 2 \pi (k/2) (t - t_p/2)^2}\]

Parameters:

• tp (float) -- Pulse length

• BW (float) -- Bandwidth of pulse

Returns:

tuple containing:

t (numpy.ndarray): Time axis

pulse (numpy.ndarray): Pulse shape

Return type:

tuple

spinlab.math.pulses.gaussian(tp, sigmas, resolution=1e-09)

Gaussian pulse

\[e^{- \frac{1}{2} \left( \frac{t - t_p/2}{\sigma} \right)^2}\]

Parameters:

• tp (float) -- Pulse length

• sigmas (float) -- Number of standard deviations where pulse is truncated

Returns:

tuple containing:

t (numpy.ndarray): Time axis

pulse (numpy.ndarray): Pulse shape

Return type:

tuple

spinlab.math.pulses.load_shape(filename)

Load a pulse shape from csv file

Parameters:

filename (str) -- Path to file

Returns:

Array of pulse shape

Return type:

numpy.ndarray

spinlab.math.pulses.plane_wave(tp, f, resolution=1e-09)

Complex plane wave pulse shape

\[e^{i 2 \pi f \left( t - \frac{t_p}{2} \right) }\]

Parameters:

• tp (float) -- Pulse length

• f (float) -- Frequency of plane wave

Returns:

tuple containing:

t (numpy.ndarray): Time axis

pulse (numpy.ndarray): Pulse shape

Return type:

tuple

spinlab.math.pulses.save_shape(pulse_shape, filename, num=101)

Save a numpy array as csv format compatible with Xepr

Parameters:

• pulse_shape (numpy.ndarray) -- Array of pulse shape

• filename (str) -- Filename to save pulse shape

spinlab.math.pulses.sinc(tp, n, resolution=1e-09)

Sinc pulse

\[ \begin{align}\begin{aligned}\frac{\sin \left( \frac{\pi}{2} (n + 1) x \right) }{x}\\x = \frac{t-\frac{t_p}{2}}{\frac{t_p}{2}}\end{aligned}\end{align} \]

Parameters:

• tp (float) -- Pulse length

• n (float) -- Total sinc lobes, must be odd for full sinc

n is number of sinc lobes, should be odd for full sinc

Returns:

tuple containing:

t (numpy.ndarray): Time axis

pulse (numpy.ndarray): Pulse shape

Return type:

tuple

spinlab.math.pulses.square(tp, t_length=0.0, resolution=1e-09)

Square pulse

Parameters:

• tp (float) -- Pulse length

• t_length (float) -- Total length of time axis

Returns:

tuple containing:

t (numpy.ndarray): Time axis

pulse (numpy.ndarray): Pulse shape

Return type:

tuple

spinlab.math.pulses.wurst(tp, N, resolution=1e-09)

Real value WURST envelope pulse shape

\[1 - \text{abs} \left( \cos \left( \frac{\pi}{t_p} (t - \frac{t_p}{2}) + \frac{\pi}{2} \right) \right) ^N\]

Parameters:

• tp (float) -- Pulse length

• N (float) -- exponential

Returns:

tuple containing:

t (numpy.ndarray): Time axis

pulse (numpy.ndarray): Pulse shape

Return type:

tuple