146 lines
4.5 KiB
Python
146 lines
4.5 KiB
Python
from time import time
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import numpy as np
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from scipy.interpolate import interp1d
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import matplotlib.pyplot as plt
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# spectral parameter
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delta = 161e3 # in Hz
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eta = 0
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lb = 10e3 # in Hz
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# correlation time
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tau = [1e-5] # in s
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# acquisition parameter
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acq_length = 4096
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dt = 1e-6 # in s
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t_echo = [0, 5e-6, 10e-6, 20e-6, 50e-6, 100e-6, 200e-6] # all in s
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# derived parameter
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t_acq = np.arange(acq_length) * dt
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t_max = acq_length*dt + 2*max(t_echo)
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dampening = np.exp(-lb * t_acq)
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# random number generator
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seed = None
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rng = np.random.default_rng(seed)
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# number of random walkers
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num_traj = 10000
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def omega_q(delta_: float, eta_: float, theta_: float, phi_: float) -> np.ndarray:
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cos_theta = np.cos(theta_)
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sin_theta = np.sin(theta_)
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return 2 * np.pi * delta_ * (3 * cos_theta * cos_theta - 1 + eta_ * sin_theta*sin_theta * np.cos(2*phi_))
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def rotate(x_in: float, y_in: float, z_in: float, a: float) -> tuple[float, float]:
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# rotation by tetrahedral angle is a given, only second angle is free parameter
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beta = 109.45 * np.pi / 180.
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cos_beta = np.cos(beta)
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sin_beta = np.sin(beta)
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cos_alpha = np.cos(a)
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sin_alpha = np.sin(a)
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scale = np.sqrt(1 - z_in * z_in) + 1e-12
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x = x_in * cos_beta + sin_beta / scale * (x_in * z_in * cos_alpha - y_in * sin_alpha)
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y = y_in * cos_beta + sin_beta / scale * (y_in * z_in * cos_alpha + x_in * sin_alpha)
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z = z_in * cos_beta - scale * sin_beta*cos_alpha
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z = max(-1, min(1, z))
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return np.arccos(z), np.arctan2(y, x)
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for tau_i in tau:
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print(f'\nStart for tau={tau_i}')
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timesignal = np.zeros((acq_length, len(t_echo)))
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start = time()
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expected_jumps = round(t_max/tau_i)
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for i in range(num_traj):
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# draw orientation
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z_theta, z_phi, z_alpha = rng.random(3)
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theta = np.arccos(1 - 2 * z_theta)
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phi = 2 * np.pi * z_phi
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alpha = 2 * np.pi * z_alpha
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# orientation in cartesian coordinates
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x_start = np.sin(theta) * np.cos(phi)
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y_start = np.sin(theta) * np.sin(phi)
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z_start = np.cos(theta)
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# calculate orientation of tetrahedral edges and their frequencies
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orientation = np.zeros(4)
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orientation[0] = omega_q(delta, eta, theta, phi)
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orientation[1] = omega_q(delta, eta, *rotate(x_start, y_start, z_start, alpha))
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orientation[2] = omega_q(delta, eta, *rotate(x_start, y_start, z_start, alpha + 2 * np.pi / 3))
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orientation[3] = omega_q(delta, eta, *rotate(x_start, y_start, z_start, alpha + 4 * np.pi / 3))
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t_passed = 0
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t = [0]
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phase = [0]
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accumulated_phase = 0
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current_orientation = rng.choice([0, 1, 2, 3])
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while t_passed < t_max:
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# orientation until the next jump
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# always jump to a different position i -> i + {1, 2, 3}
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current_orientation += rng.choice([1, 2, 3])
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current_orientation %= 4
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current_omega = orientation[current_orientation]
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# time to next jump
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t_wait = rng.exponential(tau_i)
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t_passed += t_wait
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accumulated_phase += t_wait * current_omega
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t.append(t_passed)
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phase.append(accumulated_phase)
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# convenient interpolation to get phase at arbitrary times
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phase_interpol = interp1d(t, phase)
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for j, t_echo_j in enumerate(t_echo):
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# effect of de-phasing and re-phasing
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start_amp = -2 * phase_interpol(t_echo_j)
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# start of actual acquisition
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timesignal[:, j] += np.cos(start_amp + phase_interpol(t_acq + 2*t_echo_j)) * dampening
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if (i+1) % 200 == 0:
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elapsed = time()-start
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print(f'Step {i+1} of {num_traj}', end=' - ')
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total = num_traj * elapsed / (i+1)
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print(f'elapsed: {elapsed:.2f}s - total: {total:.2f}s - remaining: {total-elapsed:.2f}s')
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timesignal /= num_traj
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# FT to spectrum
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freq = np.fft.fftshift(np.fft.fftfreq(acq_length, dt))
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spec = np.fft.fftshift(np.fft.fft(timesignal, axis=0), axes=0).real
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spec -= spec[0]
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t_echo_strings = list(map(str, t_echo))
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# plot spectra
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fig, ax = plt.subplots()
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lines = ax.plot(freq, spec)
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ax.set_title(f'tau = {tau_i}s')
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ax.legend(lines, t_echo_strings)
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# plt.savefig(f'{tau_i}.png')
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# save time signals and spectra
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# np.savetxt(f'spectrum_{tau_i}.dat', np.c_[freq, spec], header='f\t' + '\t'.join(t_echo_strings))
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# np.savetxt(f'timesignal_{tau_i}.dat', np.c_[t_acq, timesignal], header='t\t' + '\t'.join(t_echo_strings))
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plt.show()
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