# -*- coding: utf-8 -*-
"""
Created on Mon Sep 28 11:35:57 2015
@author: Stephen Jesse, Anton Ievlev, Chris R. Smith
"""
from __future__ import division, print_function, absolute_import, unicode_literals
import numpy as np
from numpy import exp, abs, sqrt, sum, real, imag, arctan2, append
from scipy.optimize import minimize
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def SHOfunc(parms, w_vec):
"""
Generates the SHO response over the given frequency band
Parameters
-----------
parms : list or tuple
SHO parae=(A,w0,Q,phi)
w_vec : 1D numpy array
Vector of frequency values
"""
return parms[0] * exp(1j * parms[3]) * parms[1] ** 2 / \
(w_vec ** 2 - 1j * w_vec * parms[1] / parms[2] - parms[1] ** 2)
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def SHOfit(parms, w_vec, resp_vec):
"""
Cost function minimization of SHO fitting
Parameters
-----------
parms : list or tuple
SHO parameters=(A,w0,Q,phi)
w_vec : 1D numpy array
Vector of frequency values
resp_vec : 1D complex numpy array or list
Cantilever response vector as a function of frequency
"""
# Cost function to minimize.
cost = lambda p: np.sum((np.abs(SHOfunc(p, w_vec)) - np.abs(resp_vec)) ** 2)
popt = minimize(cost, parms, method='TNC', options={'disp':False})
return popt.x
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def SHOestimateGuess(resp_vec, w_vec, num_points=5):
"""
Generates good initial guesses for fitting
Parameters
------------
w_vec : 1D numpy array or list
Vector of BE frequencies
resp_vec : 1D complex numpy array or list
BE response vector as a function of frequency
num_points : (Optional) unsigned int
Quality factor of the SHO peak
Returns
---------
retval : tuple
SHO fit parameters arranged as amplitude, frequency, quality factor, phase
"""
ii = np.argsort(abs(resp_vec))[::-1]
a_mat = np.array([])
e_vec = np.array([])
for c1 in range(num_points):
for c2 in range(c1 + 1, num_points):
w1 = w_vec[ii[c1]]
w2 = w_vec[ii[c2]]
X1 = real(resp_vec[ii[c1]])
X2 = real(resp_vec[ii[c2]])
Y1 = imag(resp_vec[ii[c1]])
Y2 = imag(resp_vec[ii[c2]])
denom = (w1 * (X1 ** 2 - X1 * X2 + Y1 * (Y1 - Y2)) + w2 * (-X1 * X2 + X2 ** 2 - Y1 * Y2 + Y2 ** 2))
if denom > 0:
a = ((w1 ** 2 - w2 ** 2) * (w1 * X2 * (X1 ** 2 + Y1 ** 2) - w2 * X1 * (X2 ** 2 + Y2 ** 2))) / denom
b = ((w1 ** 2 - w2 ** 2) * (w1 * Y2 * (X1 ** 2 + Y1 ** 2) - w2 * Y1 * (X2 ** 2 + Y2 ** 2))) / denom
c = ((w1 ** 2 - w2 ** 2) * (X2 * Y1 - X1 * Y2)) / denom
d = (w1 ** 3 * (X1 ** 2 + Y1 ** 2) -
w1 ** 2 * w2 * (X1 * X2 + Y1 * Y2) -
w1 * w2 ** 2 * (X1 * X2 + Y1 * Y2) +
w2 ** 3 * (X2 ** 2 + Y2 ** 2)) / denom
if d > 0:
a_mat = append(a_mat, [a, b, c, d])
A_fit = abs(a + 1j * b) / d
w0_fit = sqrt(d)
Q_fit = -sqrt(d) / c
phi_fit = arctan2(-b, -a)
H_fit = A_fit * w0_fit ** 2 * exp(1j * phi_fit) / (
w_vec ** 2 - 1j * w_vec * w0_fit / Q_fit - w0_fit ** 2)
e_vec = append(e_vec,
sum((real(H_fit) - real(resp_vec)) ** 2) +
sum((imag(H_fit) - imag(resp_vec)) ** 2))
if a_mat.size > 0:
a_mat = a_mat.reshape(-1, 4)
weight_vec = (1 / e_vec) ** 4
w_sum = sum(weight_vec)
a_w = sum(weight_vec * a_mat[:, 0]) / w_sum
b_w = sum(weight_vec * a_mat[:, 1]) / w_sum
c_w = sum(weight_vec * a_mat[:, 2]) / w_sum
d_w = sum(weight_vec * a_mat[:, 3]) / w_sum
A_fit = abs(a_w + 1j * b_w) / d_w
w0_fit = sqrt(d_w)
Q_fit = -sqrt(d_w) / c_w
phi_fit = np.arctan2(-b_w, -a_w)
H_fit = A_fit * w0_fit ** 2 * exp(1j * phi_fit) / (w_vec ** 2 - 1j * w_vec * w0_fit / Q_fit - w0_fit ** 2)
if np.std(abs(resp_vec)) / np.std(abs(resp_vec - H_fit)) < 1.2 or w0_fit < np.min(w_vec) or w0_fit > np.max(
w_vec):
p0 = SHOfastGuess(w_vec, resp_vec)
else:
p0 = np.array([A_fit, w0_fit, Q_fit, phi_fit])
else:
p0 = SHOfastGuess(w_vec, resp_vec)
return p0
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def SHOfastGuess(w_vec, resp_vec, qual_factor=200):
"""
Default SHO guess from the maximum value of the response
Parameters
------------
w_vec : 1D numpy array or list
Vector of BE frequencies
resp_vec : 1D complex numpy array or list
BE response vector as a function of frequency
qual_factor : float
Quality factor of the SHO peak
Returns
-------
retval : 1D numpy array
SHO fit parameters arranged as [amplitude, frequency, quality factor, phase]
"""
amp_vec = abs(resp_vec)
i_max = int(len(resp_vec) / 2)
return np.array([np.mean(amp_vec) / qual_factor, w_vec[i_max], qual_factor, np.angle(resp_vec[i_max])])
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def SHOlowerBound(w_vec):
"""
Provides the lower bound for the SHO fitting function
Parameters
----------
w_vec : 1D numpy array or list
Vector of BE frequencies
Returns
-------
retval : tuple
SHO fit parameters arranged as amplitude, frequency, quality factor, phase
"""
return 0, np.min(w_vec), -1e5, -np.pi
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def SHOupperBound(w_vec):
"""
Provides the upper bound for the SHO fitting function
Parameters
----------
w_vec: 1D numpy array or list
Vector of BE frequencies
Returns
-------
retval : tuple
SHO fit parameters arranged as amplitude, frequency, quality factor, phase
"""
return 1e5, np.max(w_vec), 1e5, np.pi