1 # calibcant - tools for thermally calibrating AFM cantilevers
3 # Copyright (C) 2008-2012 W. Trevor King <wking@drexel.edu>
5 # This file is part of calibcant.
7 # calibcant is free software: you can redistribute it and/or modify it under
8 # the terms of the GNU General Public License as published by the Free Software
9 # Foundation, either version 3 of the License, or (at your option) any later
12 # calibcant is distributed in the hope that it will be useful, but WITHOUT ANY
13 # WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
14 # A PARTICULAR PURPOSE. See the GNU General Public License for more details.
16 # You should have received a copy of the GNU General Public License along with
17 # calibcant. If not, see <http://www.gnu.org/licenses/>.
19 """Thermal vibration analysis.
21 Separate the more general `analyze()` from the other `vibration`
22 functions in calibcant.
24 The relevent physical quantities are :
25 Vphoto The photodiode vertical deflection voltage (what we measure)
28 >>> from pprint import pprint
32 >>> from .config import VibrationConfig
33 >>> from h5config.storage.hdf5 import pprint_HDF5
34 >>> from pypiezo.test import get_piezo_config
35 >>> from pypiezo.base import convert_volts_to_bits
37 >>> fd,filename = tempfile.mkstemp(suffix='.h5', prefix='calibcant-')
40 >>> piezo_config = get_piezo_config()
41 >>> config = VibrationConfig()
42 >>> config['frequency'] = 50e3
44 We'll be generating a test vibration time series with the following
45 parameters. Make sure these are all floats to avoid accidental
46 integer division in later steps.
49 >>> gamma = 1.6e-6 # N*s/m
51 >>> T = 1/config['frequency']
52 >>> T # doctest: +ELLIPSIS
54 >>> N = int(2**15) # count
57 where `T` is the sampling period, `N` is the number of samples, and
58 `F_sigma` is the standard deviation of the white-noise external force.
59 Note that the resonant frequency is less than the Nyquist frequency so
60 we don't have to worry too much about aliasing.
62 >>> w0 = numpy.sqrt(k/m)
63 >>> f0 = w0/(2*numpy.pi)
64 >>> f0 # doctest: +ELLIPSIS
66 >>> f_nyquist = config['frequency']/2
67 >>> f_nyquist # doctest: +ELLIPSIS
72 >>> damping = gamma / (2*m*w0)
73 >>> damping # doctest: +ELLIPSIS
79 >>> Q # doctest: +ELLIPSIS
81 >>> (1 / (2*damping)) / Q # doctest: +ELLIPSIS
84 We expect the white-noise power spectral density (PSD) to be a flat
87 >>> F0 = F_sigma**2 * 2 * T
89 because the integral from `0` `1/2T` should be `F_sigma**2`.
91 The expected time series PSD parameters are
94 >>> B = gamma/(m*2*numpy.pi)
95 >>> C = F0/(m**2*(2*numpy.pi)**4)
97 Simulate a time series with the proper PSD using center-differencing.
99 m\ddot{x} + \gamma \dot{x} + kx = F
101 m \frac{x_{i+1} - 2x_i + x_{i-1}}{T**2}
102 + \gamma \frac{x_{i+1}-x_{i-1}}{T}
105 a x_{i+1} + b x_{i} + c x_{i-1} = F_i
107 where `T` is the sampling period, `i=t/T` is the measurement index,
108 `a=m/T**2+gamma/2T`, `b=-2m/T**2+k`, and `c=m/T**2-gamma/2T`.
109 Rearranging and shifting to `j=i-1`
111 x_j = \frac{F_{i-1} - bx_{i-1} - cx_{i-2}}{a}
113 >>> a = m/T**2 + gamma/(2*T)
114 >>> b = -2*m/T**2 + k
115 >>> c = m/T**2 - gamma/(2*T)
116 >>> x = numpy.zeros((N+2,), dtype=numpy.float) # two extra initial points
117 >>> F = numpy.zeros((N,), dtype=numpy.float)
118 >>> for i in range(2, x.size):
119 ... Fp = random.gauss(mu=0, sigma=F_sigma)
123 ... x[i] = (Fp - b*xp - c*xpp)/a
124 >>> x = x[2:] # drop the initial points
126 Convert the simulated data to bits.
129 >>> deflection_bits = convert_volts_to_bits(
130 ... piezo_config.select_config('inputs', 'deflection'), x)
132 Analyze the simulated data.
134 >>> naive = analyze_naive(deflection)
135 >>> naive # doctest: +SKIP
137 >>> abs(naive / 136.9e6 - 1) < 0.1
140 >>> processed = analyze(
141 ... deflection_bits, config,
142 ... piezo_config.select_config('inputs', 'deflection'))
143 >>> processed # doctest: +SKIP
146 >>> plot(deflection=deflection_bits, config=config)
147 >>> save(filename=filename, group='/vibration/',
148 ... raw=deflection_bits, config=config,
149 ... deflection_channel_config=piezo_config.select_config(
150 ... 'inputs', 'deflection'),
151 ... processed=processed)
153 >>> pprint_HDF5(filename) # doctest: +ELLIPSIS, +REPORT_UDIFF
157 /vibration/config/deflection
158 <HDF5 dataset "analog-reference": shape (), type "|S6">
160 <HDF5 dataset "channel": shape (), type "<i4">
162 <HDF5 dataset "conversion-coefficients": shape (2,), type "<i4">
164 <HDF5 dataset "conversion-origin": shape (), type "<i4">
166 <HDF5 dataset "device": shape (), type "|S12">
168 <HDF5 dataset "inverse-conversion-coefficients": shape (2,), type "<i4">
170 <HDF5 dataset "inverse-conversion-origin": shape (), type "<i4">
172 <HDF5 dataset "maxdata": shape (), type "<i4">
174 <HDF5 dataset "name": shape (), type "|S10">
176 <HDF5 dataset "range": shape (), type "<i4">
178 <HDF5 dataset "subdevice": shape (), type "<i4">
180 /vibration/config/vibration
181 <HDF5 dataset "chunk-size": shape (), type "<i4">
183 <HDF5 dataset "frequency": shape (), type "<f8">
185 <HDF5 dataset "maximum-fit-frequency": shape (), type "<f8">
187 <HDF5 dataset "minimum-fit-frequency": shape (), type "<f8">
189 <HDF5 dataset "model": shape (), type "|S12">
191 <HDF5 dataset "overlap": shape (), type "|b1">
193 <HDF5 dataset "sample-time": shape (), type "<i4">
195 <HDF5 dataset "window": shape (), type "|S4">
198 <HDF5 dataset "data": shape (), type "<f8">
200 <HDF5 dataset "units": shape (), type "|S6">
203 <HDF5 dataset "data": shape (32768,), type "<f8">
205 <HDF5 dataset "units": shape (), type "|S4">
208 >>> data = load(filename=filename, group='/vibration/')
210 >>> pprint(data) # doctest: +ELLIPSIS
211 {'config': {'vibration': <InputChannelConfig ...>},
214 >>> data['processed'] # doctest: +SKIP
216 >>> abs(data['processed'] / 136.5e6 - 1) < 0.1
219 >>> os.remove(filename)
226 import numpy as _numpy
227 from scipy.optimize import leastsq as _leastsq
230 import matplotlib as _matplotlib
231 import matplotlib.pyplot as _matplotlib_pyplot
232 import time as _time # for timestamping lines on plots
233 except (ImportError, RuntimeError), e:
235 _matplotlib_import_error = e
237 from h5config.storage.hdf5 import HDF5_Storage as _HDF5_Storage
238 from h5config.storage.hdf5 import h5_create_group as _h5_create_group
239 import FFT_tools as _FFT_tools
240 from pypiezo.base import convert_bits_to_volts as _convert_bits_to_volts
241 from pypiezo.config import InputChannelConfig as _InputChannelConfig
243 from . import LOG as _LOG
244 from . import package_config as _package_config
245 from .config import Variance as _Variance
246 from .config import BreitWigner as _BreitWigner
247 from .config import OffsetBreitWigner as _OffsetBreitWigner
248 from .config import VibrationConfig as _VibrationConfig
249 from .util import SaveSpec as _SaveSpec
250 from .util import save as _save
251 from .util import load as _load
254 def analyze_naive(deflection):
255 """Calculate the deflection variance in Volts**2.
257 This method is simple and easy to understand, but it highly
258 succeptible to noise, drift, etc.
261 deflection : numpy array with deflection timeseries in Volts.
263 std = deflection.std()
265 _LOG.debug('naive deflection variance: %g V**2' % var)
268 def analyze(deflection, config, deflection_channel_config,
270 """Calculate the deflection variance in Volts**2.
272 Improves on `analyze_naive()` by first converting `Vphoto(t)`
273 to frequency space, and fitting a Breit-Wigner in the relevant
274 frequency range (see cantilever_calib.pdf for derivation).
275 However, there may be cases where the fit is thrown off by noise
276 spikes in frequency space. To protect from errors, the fitted
277 variance is compared to the naive variance (where all noise is
278 included), and the minimum variance is returned.
281 deflection Vphoto deflection input in bits.
282 config `.config.VibrationConfig` instance
283 deflection_channel_config
284 deflection `pypiezo.config.ChannelConfig` instance
285 plot boolean overriding matplotlib config setting.
287 The conversion to frequency space generates an average power
288 spectrum by breaking the data into windowed chunks and averaging
289 the power spectrums for the chunks together. See
290 `FFT_tools.unitary_avg_power_spectrum()` for details.
292 # convert the data from bits to volts
293 deflection_v = _convert_bits_to_volts(
294 deflection_channel_config, deflection)
295 mean = deflection_v.mean()
296 _LOG.debug('average thermal deflection (Volts): %g' % mean)
298 naive_variance = analyze_naive(deflection_v)
299 if config['model'] == _Variance:
300 return naive_variance
302 # Compute the average power spectral density per unit time (in uV**2/Hz)
303 _LOG.debug('compute the averaged power spectral density in uV**2/Hz')
304 freq_axis,power = _FFT_tools.unitary_avg_power_spectrum(
305 (deflection_v - mean)*1e6, config['frequency'],
306 config['chunk-size'], config['overlap'],
311 min_frequency=config['minimum-fit-frequency'],
312 max_frequency=config['maximum-fit-frequency'],
313 offset=config['model'] == _OffsetBreitWigner)
315 _LOG.debug('fit PSD(f) = C / ((A**2-f**2)**2 + (f*B)**2) with '
316 'A = %g, B = %g, C = %g, D = %g' % (A, B, C, D))
318 if plot or _package_config['matplotlib']:
319 _plot(deflection, freq_axis, power, A, B, C, D, config=config)
321 # Our A is in uV**2, so convert back to Volts**2
322 fitted_variance = breit_wigner_area(A,B,C) * 1e-12
324 _LOG.debug('fitted deflection variance: %g V**2' % fitted_variance)
326 if _package_config['matplotlib']:
327 plot(deflection, freq_axis, power, A, B, C, D,
330 return min(fitted_variance, naive_variance)
332 def breit_wigner(f, A, B, C, D=0):
333 """Breit-Wigner (sortof).
340 D Optional white-noise offset
342 All parameters must be postive.
344 return abs(C) / ((A**2-f**2)**2 + (B*f)**2) + abs(D)
346 def fit_psd(freq_axis, psd_data, min_frequency=500, max_frequency=25000,
348 """Fit the FFTed vibration data to a Breit-Wigner.
351 freq_axis array of frequencies in Hz
352 psd_data array of PSD amplitudes for the frequencies in freq_axis
353 min_frequency lower bound of Breit-Wigner fitting region
354 max_frequency upper bound of Breit-Wigner fitting region
355 offset add a white-noise offset to the Breit-Wigner model
357 Breit-Wigner model fit parameters `A`, `B`, `C`, and `D`.
359 # cut out the relevent frequency range
360 _LOG.debug('cut the frequency range %g to %g Hz'
361 % (min_frequency, max_frequency))
363 while freq_axis[imin] < min_frequency : imin += 1
365 while freq_axis[imax] < max_frequency : imax += 1
366 assert imax >= imin + 10 , 'less than 10 points in freq range (%g,%g)' % (
367 min_frequency, max_frequency)
369 # generate guesses for Breit-Wigner parameters A, B, C, and D
370 max_psd_index = _numpy.argmax(psd_data[imin:imax]) + imin
371 max_psd = psd_data[max_psd_index]
372 res_freq = freq_axis[max_psd_index]
374 # Breit-Wigner L(x) = C / ((A**2-x**2)**2 + (B*x)**2)
375 # is expected power spectrum for
376 # x'' + B x' + A^2 x'' = F_external(t)/m
378 # C = (2 k_B T B) / (pi m)
380 # A = resonant frequency
381 # peak at x_res = sqrt(A^2 - B^2/2) (by differentiating)
382 # which could be complex if there isn't a peak (overdamped)
383 # peak height = C / (3 x_res^4 + A^4)
387 # Guessing Q = 1 is pretty safe.
391 # so x_res^2 = B^2 Q^2 - B^2/2 = (Q^2-1/2)B^2
392 # B = x_res / sqrt(Q^2-1/2)
393 B_guess = res_freq / _numpy.sqrt(Q_guess**2-0.5)
394 A_guess = Q_guess*B_guess
395 C_guess = max_psd * (-res_freq**4 + A_guess**4)
397 D_guess = psd_data[max_psd_index]
401 _LOG.debug(('guessed params: resonant freq %g, max psd %g, Q %g, A %g, '
402 'B %g, C %g, D %g') % (
403 res_freq, max_psd, Q_guess, A_guess, B_guess, C_guess, D_guess))
404 # Half width w on lower side when L(a-w) = L(a)/2
405 # (a**2 - (a-w)**2)**2 + (b*(a-w))**2 = 2*(b*a)**2
406 # Let W=(a-w)**2, A=a**2, and B=b**2
407 # (A - W)**2 + BW = 2*AB
408 # W**2 - 2AW + A**2 + BW = 2AB
409 # W**2 + (B-2A)W + (A**2-2AB) = 0
410 # W = (2A-B)/2 * [1 +/- sqrt(1 - 4(A**2-2AB)/(B-2A)**2]
411 # = (2A-B)/2 * [1 +/- sqrt(1 - 4A(A-2B)/(B-2A)**2]
412 # (a-w)**2 = (2A-B)/2 * [1 +/- sqrt(1 - 4A(A-2B)/(B-2A)**2]
413 # so w is a disaster ;)
414 # For some values of A and B (non-underdamped), W is imaginary or negative.
416 # The mass m is given by m = k_B T / (pi A)
417 # The spring constant k is given by k = m (omega_0)**2
418 # The quality factor Q is given by Q = omega_0 m / gamma
420 # Fitting the PSD of V = photoSensitivity*x just rescales the parameters
422 # fit Breit-Wigner using scipy.optimize.leastsq
423 def residual(p, y, x):
424 return breit_wigner(x, *p) - y
426 guess = _numpy.array((A_guess, B_guess, C_guess, D_guess))
428 guess = _numpy.array((A_guess, B_guess, C_guess))
430 p,cov,info,mesg,ier = _leastsq(
432 args=(psd_data[imin:imax], freq_axis[imin:imax]),
433 full_output=True, maxfev=10000)
434 _LOG.debug('fitted params: %s' % p)
435 _LOG.debug('covariance matrix: %s' % cov)
436 #_LOG.debug('info: %s' % info)
437 _LOG.debug('message: %s' % mesg)
439 _LOG.debug('solution converged')
441 _LOG.debug('solution did not converge')
447 A=abs(A) # A and B only show up as squares in f(x)
448 B=abs(B) # so ensure we get positive values.
449 C=abs(C) # Only abs(C) is used in breit_wigner().
452 def breit_wigner_area(A, B, C):
453 # Integrating the the power spectral density per unit time (power) over the
454 # frequency axis [0, fN] returns the total signal power per unit time
455 # int_0^fN power(f)df = 1/T int_0^T |x(t)**2| dt
456 # = <V(t)**2>, the variance for our AC signal.
457 # The variance from our fitted Breit-Wigner is the area under the Breit-Wigner
458 # <V(t)**2> = (pi*C) / (2*B*A**2)
459 return (_numpy.pi * C) / (2 * B * A**2)
461 def save(filename, group='/', raw=None, config=None,
462 deflection_channel_config=None, processed=None):
464 _SaveSpec(item=config, relpath='config/vibration',
465 config=_VibrationConfig),
466 _SaveSpec(item=deflection_channel_config,
467 relpath='config/deflection',
468 config=_InputChannelConfig),
469 _SaveSpec(item=raw, relpath='raw', units='bits'),
470 _SaveSpec(item=processed, relpath='processed', units='V^2/Hz'),
472 _save(filename=filename, group=group, specs=specs)
474 def load(filename=None, group='/'):
476 _SaveSpec(key=('config', 'vibration'), relpath='config/vibration',
477 config=_VibrationConfig),
478 _SaveSpec(key=('config', 'deflection'), relpath='config/deflection',
479 config=_InputChannelConfig),
480 _SaveSpec(key=('raw',), relpath='raw', array=True, units='bits'),
481 _SaveSpec(key=('processed',), relpath='processed', units='V^2/Hz'),
483 return _load(filename=filename, group=group, specs=specs)
485 def plot(deflection=None, freq_axis=None, power=None, A=None, B=None,
486 C=None, D=0, config=None, analyze=False):
487 """Plot 3 subfigures displaying vibration data and analysis.
489 Time series (Vphoto vs sample index) (show any major drift events),
490 A histogram of Vphoto, with a gaussian fit (demonstrate randomness), and
491 FFTed Vphoto data (Vphoto vs Frequency) (show major noise components).
494 raise _matplotlib_import_error
495 figure = _matplotlib_pyplot.figure()
498 assert deflection != None, (
499 'must set at least one of `deflection` or `power`.')
500 time_axes = figure.add_subplot(2, 1, 1)
501 hist_axes = figure.add_subplot(2, 1, 2)
503 elif deflection is None:
504 time_axes = hist_axes = None
505 freq_axes = figure.add_subplot(1, 1, 1)
507 time_axes = figure.add_subplot(3, 1, 1)
508 hist_axes = figure.add_subplot(3, 1, 2)
509 freq_axes = figure.add_subplot(3, 1, 3)
511 timestamp = _time.strftime('%H%M%S')
513 if deflection is not None:
514 time_axes.plot(deflection, 'r.')
515 time_axes.autoscale(tight=True)
516 time_axes.set_title('free oscillation')
518 # plot histogram distribution and gaussian fit
520 n,bins,patches = hist_axes.hist(
521 deflection, bins=30, normed=True, align='mid')
522 gauss = _numpy.zeros((len(bins),), dtype=_numpy.float)
523 mean = deflection.mean()
524 std = deflection.std()
527 gauss = _numpy.sqrt(2*pi)/std * exp(-0.5*((bins-mean)/std)**2)
528 # Matplotlib's normed histogram uses bin heights of n/(len(x)*dbin)
529 dbin = bins[1]-bins[0]
530 hist_axes.plot(bins, gauss/dbin, 'r-')
531 hist_axes.autoscale(tight=True)
532 if power is not None:
534 freq_axes.set_yscale('log')
535 freq_axes.plot(freq_axis, power, 'r.-')
536 freq_axes.autoscale(tight=True)
537 xmin,xmax = freq_axes.get_xbound()
538 ymin,ymax = freq_axes.get_ybound()
540 # highlight the region we're fitting
542 config['minimum-fit-frequency'],
543 config['maximum-fit-frequency'],
544 facecolor='g', alpha=0.1, zorder=-2)
547 fitdata = breit_wigner(freq_axis, A, B, C, D)
548 freq_axes.plot(freq_axis, fitdata, 'b-')
549 noisefloor = D + 0*freq_axis;
550 freq_axes.plot(freq_axis, noisefloor)
553 res_freq = _numpy.sqrt(A**2 - B**2/2)
554 freq_axes.axvline(res_freq, color='b', zorder=-1)
556 freq_axes.set_title('power spectral density %s' % timestamp)
557 freq_axes.axis([xmin,xmax,ymin,ymax])
558 freq_axes.set_xlabel('frequency (Hz)')
559 freq_axes.set_ylabel('PSD')
560 if hasattr(figure, 'show'):
562 _plot = plot # alternative name for use inside analyze()