1 # calibcant - tools for thermally calibrating AFM cantilevers
3 # Copyright (C) 2008-2011 W. Trevor King <wking@drexel.edu>
5 # This file is part of calibcant.
7 # calibcant is free software: you can redistribute it and/or
8 # modify it under the terms of the GNU Lesser General Public
9 # License as published by the Free Software Foundation, either
10 # version 3 of the License, or (at your option) any later version.
12 # calibcant is distributed in the hope that it will be useful,
13 # but WITHOUT ANY WARRANTY; without even the implied warranty of
14 # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
15 # GNU Lesser General Public License for more details.
17 # You should have received a copy of the GNU Lesser General Public
18 # License along with calibcant. If not, see
19 # <http://www.gnu.org/licenses/>.
21 """Thermal vibration analysis.
23 Separate the more general `vib_analyze()` from the other `vib_*()`
24 functions in calibcant.
26 The relevent physical quantities are :
27 Vphoto The photodiode vertical deflection voltage (what we measure)
30 >>> from pprint import pprint
34 >>> from .config import VibrationConfig
35 >>> from h5config.storage.hdf5 import pprint_HDF5
36 >>> from pypiezo.test import get_piezo_config
37 >>> from pypiezo.base import convert_volts_to_bits
39 >>> fd,filename = tempfile.mkstemp(suffix='.h5', prefix='calibcant-')
42 >>> piezo_config = get_piezo_config()
43 >>> vibration_config = VibrationConfig()
44 >>> vibration_config['frequency'] = 50e3
46 We'll be generating a test vibration time series with the following
47 parameters. Make sure these are all floats to avoid accidental
48 integer division in later steps.
51 >>> gamma = 1.6e-6 # N*s/m
53 >>> T = 1/vibration_config['frequency']
54 >>> T # doctest: +ELLIPSIS
56 >>> N = int(2**15) # count
59 where `T` is the sampling period, `N` is the number of samples, and
60 `F_sigma` is the standard deviation of the white-noise external force.
61 Note that the resonant frequency is less than the Nyquist frequency so
62 we don't have to worry too much about aliasing.
64 >>> w0 = numpy.sqrt(k/m)
65 >>> f0 = w0/(2*numpy.pi)
66 >>> f0 # doctest: +ELLIPSIS
68 >>> f_nyquist = vibration_config['frequency']/2
69 >>> f_nyquist # doctest: +ELLIPSIS
74 >>> damping = gamma / (2*m*w0)
75 >>> damping # doctest: +ELLIPSIS
81 >>> Q # doctest: +ELLIPSIS
83 >>> (1 / (2*damping)) / Q # doctest: +ELLIPSIS
86 We expect the white-noise power spectral density (PSD) to be a flat
89 >>> F0 = F_sigma**2 * 2 * T
91 because the integral from `0` `1/2T` should be `F_sigma**2`.
93 The expected time series PSD parameters are
96 >>> B = gamma/(m*2*numpy.pi)
97 >>> C = F0/(m**2*(2*numpy.pi)**4)
99 Simulate a time series with the proper PSD using center-differencing.
101 m\ddot{x} + \gamma \dot{x} + kx = F
103 m \frac{x_{i+1} - 2x_i + x_{i-1}}{T**2}
104 + \gamma \frac{x_{i+1}-x_{i-1}}{T}
107 a x_{i+1} + b x_{i} + c x_{i-1} = F_i
109 where `T` is the sampling period, `i=t/T` is the measurement index,
110 `a=m/T**2+gamma/2T`, `b=-2m/T**2+k`, and `c=m/T**2-gamma/2T`.
111 Rearranging and shifting to `j=i-1`
113 x_j = \frac{F_{i-1} - bx_{i-1} - cx_{i-2}}{a}
115 >>> a = m/T**2 + gamma/(2*T)
116 >>> b = -2*m/T**2 + k
117 >>> c = m/T**2 - gamma/(2*T)
118 >>> x = numpy.zeros((N+2,), dtype=numpy.float) # two extra initial points
119 >>> F = numpy.zeros((N,), dtype=numpy.float)
120 >>> for i in range(2, x.size):
121 ... Fp = random.gauss(mu=0, sigma=F_sigma)
125 ... x[i] = (Fp - b*xp - c*xpp)/a
126 >>> x = x[2:] # drop the initial points
128 Convert the simulated data to bits.
131 >>> deflection_bits = convert_volts_to_bits(
132 ... piezo_config.select_config('inputs', 'deflection'), x)
134 Analyze the simulated data.
136 >>> naive_vibration = vib_analyze_naive(deflection)
137 >>> naive_vibration # doctest: +SKIP
139 >>> abs(naive_vibration / 136.9e6 - 1) < 0.1
142 >>> processed_vibration = vib_analyze(
143 ... deflection_bits, vibration_config,
144 ... piezo_config.select_config('inputs', 'deflection'))
145 >>> processed_vibration # doctest: +SKIP
148 >>> vib_plot(deflection=deflection_bits, vibration_config=vibration_config)
149 >>> vib_save(filename=filename, group='/vibration/',
150 ... raw_vibration=deflection_bits, vibration_config=vibration_config,
151 ... deflection_channel_config=piezo_config.select_config(
152 ... 'inputs', 'deflection'),
153 ... processed_vibration=processed_vibration)
155 >>> pprint_HDF5(filename) # doctest: +ELLIPSIS, +REPORT_UDIFF
159 /vibration/config/deflection
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">
197 <HDF5 dataset "processed": shape (), type "<f8">
200 <HDF5 dataset "deflection": shape (32768,), type "<f8">
203 >>> (raw_vibration,vibration_config,deflection_channel_config,
204 ... processed_vibration) = vib_load(
205 ... filename=filename, group='/vibration/')
207 >>> processed_vibration # doctest: +SKIP
209 >>> abs(processed_vibration / 136.5e6 - 1) < 0.1
212 >>> os.remove(filename)
219 import numpy as _numpy
220 from scipy.optimize import leastsq as _leastsq
223 import matplotlib as _matplotlib
224 import matplotlib.pyplot as _matplotlib_pyplot
225 import time as _time # for timestamping lines on plots
226 except (ImportError, RuntimeError), e:
228 _matplotlib_import_error = e
230 from h5config.storage.hdf5 import HDF5_Storage as _HDF5_Storage
231 from h5config.storage.hdf5 import h5_create_group as _h5_create_group
232 import FFT_tools as _FFT_tools
233 from pypiezo.base import convert_bits_to_volts as _convert_bits_to_volts
234 from pypiezo.config import ChannelConfig as _ChannelConfig
236 from . import LOG as _LOG
237 from . import package_config as _package_config
238 from .config import Variance as _Variance
239 from .config import BreitWigner as _BreitWigner
240 from .config import OffsetBreitWigner as _OffsetBreitWigner
241 from .config import VibrationConfig as _VibrationConfig
244 def vib_analyze_naive(deflection):
245 """Calculate the deflection variance in Volts**2.
247 This method is simple and easy to understand, but it highly
248 succeptible to noise, drift, etc.
251 deflection : numpy array with deflection timeseries in Volts.
253 std = deflection.std()
255 _LOG.debug('naive deflection variance: %g V**2' % var)
258 def vib_analyze(deflection, vibration_config, deflection_channel_config,
260 """Calculate the deflection variance in Volts**2.
262 Improves on `vib_analyze_naive()` by first converting `Vphoto(t)`
263 to frequency space, and fitting a Breit-Wigner in the relevant
264 frequency range (see cantilever_calib.pdf for derivation).
265 However, there may be cases where the fit is thrown off by noise
266 spikes in frequency space. To protect from errors, the fitted
267 variance is compared to the naive variance (where all noise is
268 included), and the minimum variance is returned.
271 deflection Vphoto deflection input in bits.
272 vibration_config `.config._VibrationConfig` instance
273 deflection_channel_config
274 deflection `pypiezo.config.ChannelConfig` instance
275 plot boolean overriding matplotlib config setting.
277 The conversion to frequency space generates an average power
278 spectrum by breaking the data into windowed chunks and averaging
279 the power spectrums for the chunks together. See
280 `FFT_tools.unitary_avg_power_spectrum()` for details.
282 # convert the data from bits to volts
283 deflection_v = _convert_bits_to_volts(
284 deflection_channel_config, deflection)
285 mean = deflection_v.mean()
286 _LOG.debug('average thermal deflection (Volts): %g' % mean)
288 naive_variance = vib_analyze_naive(deflection_v)
289 if vibration_config['model'] == _Variance:
290 return naive_variance
292 # Compute the average power spectral density per unit time (in uV**2/Hz)
293 _LOG.debug('compute the averaged power spectral density in uV**2/Hz')
294 freq_axis,power = _FFT_tools.unitary_avg_power_spectrum(
295 (deflection_v - mean)*1e6, vibration_config['frequency'],
296 vibration_config['chunk-size'], vibration_config['overlap'],
297 vibration_config['window'])
301 min_frequency=vibration_config['minimum-fit-frequency'],
302 max_frequency=vibration_config['maximum-fit-frequency'],
303 offset=vibration_config['model'] == _OffsetBreitWigner)
305 _LOG.debug('fit PSD(f) = C / ((A**2-f**2)**2 + (f*B)**2) with '
306 'A = %g, B = %g, C = %g, D = %g' % (A, B, C, D))
308 if plot or _package_config['matplotlib']:
309 vib_plot(deflection, freq_axis, power, A, B, C, D,
310 vibration_config=vibration_config)
312 # Our A is in uV**2, so convert back to Volts**2
313 fitted_variance = breit_wigner_area(A,B,C) * 1e-12
315 _LOG.debug('fitted deflection variance: %g V**2' % fitted_variance)
317 if _package_config['matplotlib']:
318 vib_plot(deflection, freq_axis, power, A, B, C, D,
319 vibration_config=vibration_config)
321 return min(fitted_variance, naive_variance)
323 def breit_wigner(f, A, B, C, D=0):
324 """Breit-Wigner (sortof).
331 D Optional white-noise offset
333 All parameters must be postive.
335 return abs(C) / ((A**2-f**2)**2 + (B*f)**2) + abs(D)
337 def fit_psd(freq_axis, psd_data, min_frequency=500, max_frequency=25000,
339 """Fit the FFTed vibration data to a Breit-Wigner.
342 freq_axis array of frequencies in Hz
343 psd_data array of PSD amplitudes for the frequencies in freq_axis
344 min_frequency lower bound of Breit-Wigner fitting region
345 max_frequency upper bound of Breit-Wigner fitting region
346 offset add a white-noise offset to the Breit-Wigner model
348 Breit-Wigner model fit parameters `A`, `B`, `C`, and `D`.
350 # cut out the relevent frequency range
351 _LOG.debug('cut the frequency range %g to %g Hz'
352 % (min_frequency, max_frequency))
354 while freq_axis[imin] < min_frequency : imin += 1
356 while freq_axis[imax] < max_frequency : imax += 1
357 assert imax >= imin + 10 , 'less than 10 points in freq range (%g,%g)' % (
358 min_frequency, max_frequency)
360 # generate guesses for Breit-Wigner parameters A, B, C, and D
361 max_psd_index = _numpy.argmax(psd_data[imin:imax]) + imin
362 max_psd = psd_data[max_psd_index]
363 res_freq = freq_axis[max_psd_index]
365 # Breit-Wigner L(x) = C / ((A**2-x**2)**2 + (B*x)**2)
366 # is expected power spectrum for
367 # x'' + B x' + A^2 x'' = F_external(t)/m
369 # C = (2 k_B T B) / (pi m)
371 # A = resonant frequency
372 # peak at x_res = sqrt(A^2 - B^2/2) (by differentiating)
373 # which could be complex if there isn't a peak (overdamped)
374 # peak height = C / (3 x_res^4 + A^4)
378 # Guessing Q = 1 is pretty safe.
382 # so x_res^2 = B^2 Q^2 - B^2/2 = (Q^2-1/2)B^2
383 # B = x_res / sqrt(Q^2-1/2)
384 B_guess = res_freq / _numpy.sqrt(Q_guess**2-0.5)
385 A_guess = Q_guess*B_guess
386 C_guess = max_psd * (-res_freq**4 + A_guess**4)
388 D_guess = psd_data[max_psd_index]
392 _LOG.debug(('guessed params: resonant freq %g, max psd %g, Q %g, A %g, '
393 'B %g, C %g, D %g') % (
394 res_freq, max_psd, Q_guess, A_guess, B_guess, C_guess, D_guess))
395 # Half width w on lower side when L(a-w) = L(a)/2
396 # (a**2 - (a-w)**2)**2 + (b*(a-w))**2 = 2*(b*a)**2
397 # Let W=(a-w)**2, A=a**2, and B=b**2
398 # (A - W)**2 + BW = 2*AB
399 # W**2 - 2AW + A**2 + BW = 2AB
400 # W**2 + (B-2A)W + (A**2-2AB) = 0
401 # W = (2A-B)/2 * [1 +/- sqrt(1 - 4(A**2-2AB)/(B-2A)**2]
402 # = (2A-B)/2 * [1 +/- sqrt(1 - 4A(A-2B)/(B-2A)**2]
403 # (a-w)**2 = (2A-B)/2 * [1 +/- sqrt(1 - 4A(A-2B)/(B-2A)**2]
404 # so w is a disaster ;)
405 # For some values of A and B (non-underdamped), W is imaginary or negative.
407 # The mass m is given by m = k_B T / (pi A)
408 # The spring constant k is given by k = m (omega_0)**2
409 # The quality factor Q is given by Q = omega_0 m / gamma
411 # Fitting the PSD of V = photoSensitivity*x just rescales the parameters
413 # fit Breit-Wigner using scipy.optimize.leastsq
414 def residual(p, y, x):
415 return breit_wigner(x, *p) - y
417 guess = _numpy.array((A_guess, B_guess, C_guess, D_guess))
419 guess = _numpy.array((A_guess, B_guess, C_guess))
421 p,cov,info,mesg,ier = _leastsq(
423 args=(psd_data[imin:imax], freq_axis[imin:imax]),
424 full_output=True, maxfev=10000)
425 _LOG.debug('fitted params: %s' % p)
426 _LOG.debug('covariance matrix: %s' % cov)
427 #_LOG.debug('info: %s' % info)
428 _LOG.debug('message: %s' % mesg)
430 _LOG.debug('solution converged')
432 _LOG.debug('solution did not converge')
438 A=abs(A) # A and B only show up as squares in f(x)
439 B=abs(B) # so ensure we get positive values.
440 C=abs(C) # Only abs(C) is used in breit_wigner().
443 def breit_wigner_area(A, B, C):
444 # Integrating the the power spectral density per unit time (power) over the
445 # frequency axis [0, fN] returns the total signal power per unit time
446 # int_0^fN power(f)df = 1/T int_0^T |x(t)**2| dt
447 # = <V(t)**2>, the variance for our AC signal.
448 # The variance from our fitted Breit-Wigner is the area under the Breit-Wigner
449 # <V(t)**2> = (pi*C) / (2*B*A**2)
450 return (_numpy.pi * C) / (2 * B * A**2)
452 def vib_save(filename, group='/', raw_vibration=None, vibration_config=None,
453 deflection_channel_config=None, processed_vibration=None):
454 with _h5py.File(filename, 'a') as f:
455 cwg = _h5_create_group(f, group)
456 if raw_vibration is not None:
458 del cwg['raw/deflection']
461 cwg['raw/deflection'] = raw_vibration
462 storage = _HDF5_Storage()
463 for config,key in [(vibration_config, 'config/vibration'),
464 (deflection_channel_config,
465 'config/deflection')]:
468 config_cwg = _h5_create_group(cwg, key)
469 storage.save(config=config, group=config_cwg)
470 if processed_vibration is not None:
475 cwg['processed'] = processed_vibration
477 def vib_load(filename, group='/'):
478 assert group.endswith('/')
479 raw_vibration = processed_vibration = None
481 with _h5py.File(filename, 'a') as f:
483 raw_vibration = f[group+'raw/deflection'][...]
486 for Config,key in [(_VibrationConfig, 'config/vibration'),
487 (_ChannelConfig, 'config/deflection')]:
488 config = Config(storage=_HDF5_Storage(
489 filename=filename, group=group+key))
490 configs.append(config)
492 processed_vibration = f[group+'processed'][...]
495 ret = [raw_vibration]
497 ret.append(processed_vibration)
498 for config in configs:
502 def vib_plot(deflection=None, freq_axis=None, power=None, A=None, B=None,
503 C=None, D=0, vibration_config=None, analyze=False):
504 """Plot 3 subfigures displaying vibration data and analysis.
506 Time series (Vphoto vs sample index) (show any major drift events),
507 A histogram of Vphoto, with a gaussian fit (demonstrate randomness), and
508 FFTed Vphoto data (Vphoto vs Frequency) (show major noise components).
511 raise _matplotlib_import_error
512 figure = _matplotlib_pyplot.figure()
515 assert deflection != None, (
516 'must set at least one of `deflection` or `power`.')
517 time_axes = figure.add_subplot(2, 1, 1)
518 hist_axes = figure.add_subplot(2, 1, 2)
520 elif deflection is None:
521 time_axes = hist_axes = None
522 freq_axes = figure.add_subplot(1, 1, 1)
524 time_axes = figure.add_subplot(3, 1, 1)
525 hist_axes = figure.add_subplot(3, 1, 2)
526 freq_axes = figure.add_subplot(3, 1, 3)
528 timestamp = _time.strftime('%H%M%S')
530 if deflection is not None:
531 time_axes.plot(deflection, 'r.')
532 time_axes.set_title('free oscillation')
534 # plot histogram distribution and gaussian fit
536 n,bins,patches = hist_axes.hist(
537 deflection, bins=30, normed=True, align='mid')
538 gauss = _numpy.zeros((len(bins),), dtype=_numpy.float)
539 mean = deflection.mean()
540 std = deflection.std()
543 gauss = _numpy.sqrt(2*pi)/std * exp(-0.5*((bins-mean)/std)**2)
544 # Matplotlib's normed histogram uses bin heights of n/(len(x)*dbin)
545 dbin = bins[1]-bins[0]
546 hist_axes.plot(bins, gauss/dbin, 'r-')
547 if power is not None:
549 freq_axes.set_yscale('log')
550 freq_axes.plot(freq_axis, power, 'r.-')
551 xmin,xmax = freq_axes.get_xbound()
552 ymin,ymax = freq_axes.get_ybound()
554 # highlight the region we're fitting
556 vibration_config['minimum-fit-frequency'],
557 vibration_config['maximum-fit-frequency'],
558 facecolor='g', alpha=0.1, zorder=-2)
561 fitdata = breit_wigner(freq_axis, A, B, C, D)
562 freq_axes.plot(freq_axis, fitdata, 'b-')
563 noisefloor = D + 0*freq_axis;
564 freq_axes.plot(freq_axis, noisefloor)
567 res_freq = _numpy.sqrt(A**2 - B**2/2)
568 freq_axes.axvline(res_freq, color='b', zorder=-1)
570 freq_axes.set_title('power spectral density %s' % timestamp)
571 freq_axes.axis([xmin,xmax,ymin,ymax])
572 freq_axes.set_xlabel('frequency (Hz)')
573 freq_axes.set_ylabel('PSD')