# You should have received a copy of the GNU General Public License
# along with this program. If not, see <http://www.gnu.org/licenses/>.
-"""Wrap Numpy's fft module to reduce clutter.
+"""Wrap Numpy's :py:mod:`~numpy.fft` module to reduce clutter.
Provides a unitary discrete FFT and a windowed version based on
-numpy.fft.rfft.
-
-Create some fake data:
-
->>> import numpy
->>> data = numpy.random.rand(10)
->>> frequency = 1
+:py:func:`numpy.fft.rfft`.
Main entry functions:
->>> rfft = unitary_rfft(data, freq=frequency)
->>> psd = power_spectrum(data, freq=frequency)
->>> upsd = unitary_power_spectrum(data, freq=frequency)
->>> apsd = avg_power_spectrum(data, freq=frequency, chunk_size=2048,
-... overlap=True, window=window_hann)
->>> aupsd = unitary_avg_power_spectrum(data, freq=frequency, chunk_size=2048,
-... overlap=True, window=window_hann)
+* :py:func:`unitary_rfft`
+* :py:func:`power_spectrum`
+* :py:func:`unitary_power_spectrum`
+* :py:func:`avg_power_spectrum`
+* :py:func:`unitary_avg_power_spectrum`
"""
+import logging as _logging
+import unittest as _unittest
+
import numpy as _numpy
+try:
+ import matplotlib.pyplot as _pyplot
+except (ImportError, RuntimeError) as e:
+ _pyplot = None
+ _pyplot_import_error = e
+
+__version__ = '0.5'
+
+
+LOG = _logging.getLogger('FFT-tools')
+LOG.addHandler(_logging.StreamHandler())
+LOG.setLevel(_logging.ERROR)
-__version__ = '0.3'
# Display time- and freq-space plots of the test transforms if True
TEST_PLOTS = False
def floor_pow_of_two(num):
- "Round num down to the closest exact a power of two."
+ """Round `num` down to the closest exact a power of two.
+
+ Examples
+ --------
+
+ >>> floor_pow_of_two(3)
+ 2
+ >>> floor_pow_of_two(11)
+ 8
+ >>> floor_pow_of_two(15)
+ 8
+ """
lnum = _numpy.log2(num)
if int(lnum) != lnum:
num = 2**_numpy.floor(lnum)
- return num
+ return int(num)
def round_pow_of_two(num):
- "Round num to the closest exact a power of two on a log scale."
+ """Round `num` to the closest exact a power of two on a log scale.
+
+ Examples
+ --------
+
+ >>> round_pow_of_two(2.9) # Note rounding on *log scale*
+ 4
+ >>> round_pow_of_two(11)
+ 8
+ >>> round_pow_of_two(15)
+ 16
+ """
lnum = _numpy.log2(num)
if int(lnum) != lnum:
num = 2**_numpy.round(lnum)
- return num
+ return int(num)
def ceil_pow_of_two(num):
- "Round num up to the closest exact a power of two."
+ """Round `num` up to the closest exact a power of two.
+
+ Examples
+ --------
+
+ >>> ceil_pow_of_two(3)
+ 4
+ >>> ceil_pow_of_two(11)
+ 16
+ >>> ceil_pow_of_two(15)
+ 16
+ """
lnum = _numpy.log2(num)
if int(lnum) != lnum:
num = 2**_numpy.ceil(lnum)
- return num
-
-
-def _test_rfft(xs, Xs):
- # Numpy's FFT algoritm returns
- # n-1
- # X[k] = SUM x[m] exp (-j 2pi km /n)
- # m=0
- # (see http://www.tramy.us/numpybook.pdf)
- j = _numpy.complex(0,1)
- n = len(xs)
- Xa = []
- for k in range(n):
- Xa.append(sum([x*_numpy.exp(-j*2*_numpy.pi*k*m/n)
- for x,m in zip(xs,range(n))]))
- if k < len(Xs):
- if (Xs[k]-Xa[k])/_numpy.abs(Xa[k]) >= 1e-6:
- raise ValueError(
- ('rfft mismatch on element {}: {} != {}, relative error {}'
- ).format(
- k, Xs[k], Xa[k], (Xs[k]-Xa[k])/_numpy.abs(Xa[k])))
- # Which should satisfy the discrete form of Parseval's theorem
- # n-1 n-1
- # SUM |x_m|^2 = 1/n SUM |X_k|^2.
- # m=0 k=0
- timeSum = sum([_numpy.abs(x)**2 for x in xs])
- freqSum = sum([_numpy.abs(X)**2 for X in Xa])
- if _numpy.abs(freqSum/_numpy.float(n) - timeSum)/timeSum >= 1e-6:
- raise ValueError(
- "Mismatch on Parseval's, {} != 1/{} * {}".format(
- timeSum, n, freqSum))
+ return int(num)
-def _test_rfft_suite():
- print('Test numpy rfft definition')
- xs = [1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1]
- _test_rfft(xs, _numpy.fft.rfft(xs))
+def unitary_rfft(data, freq=1.0):
+ """Compute the unitary Fourier transform of real data.
+ Unitary = preserves power [Parseval's theorem].
-def unitary_rfft(data, freq=1.0):
- """Compute the Fourier transform of real data.
+ Parameters
+ ----------
+ data : iterable
+ Real (not complex) data taken with a sampling frequency `freq`.
+ freq : real
+ Sampling frequency.
- Unitary (preserves power [Parseval's theorem]).
+ Returns
+ -------
+ freq_axis,trans : numpy.ndarray
+ Arrays ready for plotting.
+ Notes
+ -----
If the units on your data are Volts,
and your sampling frequency is in Hz,
- then freq_axis will be in Hz,
- and trans will be in Volts.
+ then `freq_axis` will be in Hz,
+ and `trans` will be in Volts.
"""
nsamps = floor_pow_of_two(len(data))
# Which should satisfy the discrete form of Parseval's theorem
# continuous transformed data with (also NR 12.1.8)
# X'_k = dtX = X / <sampling freq>
trans = _numpy.fft.rfft(data[0:nsamps]) / _numpy.float(freq)
- freq_axis = _numpy.linspace(0, freq/2, nsamps/2+1)
+ freq_axis = _numpy.linspace(0, freq / 2, nsamps / 2 + 1)
return (freq_axis, trans)
-def _test_unitary_rfft_parsevals(xs, freq, freqs, Xs):
- # Which should satisfy the discretized integral form of Parseval's theorem
- # n-1 n-1
- # SUM |x_m|^2 dt = SUM |X_k|^2 df
- # m=0 k=0
- dt = 1.0/freq
- df = freqs[1]-freqs[0]
- if df - 1/(len(xs)*dt)/df >= 1e-6:
- raise ValueError(
- 'Mismatch in spacing, {} != 1/({}*{})'.format(df, len(xs), dt))
- Xa = list(Xs)
- for k in range(len(Xs)-1,1,-1):
- Xa.append(Xa[k])
- if len(xs) != len(Xa):
- raise ValueError('Length mismatch {} != {}'.format(len(xs), len(Xa)))
- lhs = sum([_numpy.abs(x)**2 for x in xs]) * dt
- rhs = sum([_numpy.abs(X)**2 for X in Xa]) * df
- if _numpy.abs(lhs - rhs)/lhs >= 1e-4:
- raise ValueError("Mismatch on Parseval's, {} != {}".format(lhs, rhs))
-
-
-def _test_unitary_rfft_parsevals_suite():
- print("Test unitary rfft on Parseval's theorem")
- xs = [1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1]
- dt = _numpy.pi
- freqs,Xs = unitary_rfft(xs, 1.0/dt)
- _test_unitary_rfft_parsevals(xs, 1.0/dt, freqs, Xs)
-
-
-def _rect(t):
- if _numpy.abs(t) < 0.5:
- return 1
- else:
- return 0
-
-
-def _test_unitary_rfft_rect(
- a=1.0, time_shift=5.0, samp_freq=25.6, samples=256):
- "Show fft(rect(at)) = 1/abs(a) * _numpy.sinc(f/a)"
- samp_freq = _numpy.float(samp_freq)
- a = _numpy.float(a)
-
- x = _numpy.zeros((samples,), dtype=_numpy.float)
- dt = 1.0/samp_freq
- for i in range(samples):
- t = i*dt
- x[i] = _rect(a*(t-time_shift))
- freq_axis, X = unitary_rfft(x, samp_freq)
- #_test_unitary_rfft_parsevals(x, samp_freq, freq_axis, X)
-
- # remove the phase due to our time shift
- j = _numpy.complex(0.0,1.0) # sqrt(-1)
- for i in range(len(freq_axis)):
- f = freq_axis[i]
- inverse_phase_shift = _numpy.exp(j*2.0*_numpy.pi*time_shift*f)
- X[i] *= inverse_phase_shift
-
- expected = _numpy.zeros((len(freq_axis),), dtype=_numpy.float)
- # normalized sinc(x) = sin(pi x)/(pi x)
- # so sinc(0.5) = sin(pi/2)/(pi/2) = 2/pi
- if _numpy.sinc(0.5) != 2.0/_numpy.pi:
- raise ValueError('abnormal sinc()')
- for i in range(len(freq_axis)):
- f = freq_axis[i]
- expected[i] = 1.0/_numpy.abs(a) * _numpy.sinc(f/a)
-
- if TEST_PLOTS:
- figure = _pyplot.figure()
- time_axes = figure.add_subplot(2, 1, 1)
- time_axes.plot(_numpy.arange(0, dt*samples, dt), x)
- time_axes.set_title('time series')
- freq_axes = figure.add_subplot(2, 1, 2)
- freq_axes.plot(freq_axis, X.real, 'r.')
- freq_axes.plot(freq_axis, X.imag, 'g.')
- freq_axes.plot(freq_axis, expected, 'b-')
- freq_axes.set_title('freq series')
-
-
-def _test_unitary_rfft_rect_suite():
- print('Test unitary FFTs on variously shaped rectangular functions')
- _test_unitary_rfft_rect(a=0.5)
- _test_unitary_rfft_rect(a=2.0)
- _test_unitary_rfft_rect(a=0.7, samp_freq=50, samples=512)
- _test_unitary_rfft_rect(a=3.0, samp_freq=60, samples=1024)
-
-
-def _gaussian(a, t):
- return _numpy.exp(-a * t**2)
-
-
-def _test_unitary_rfft_gaussian(
- a=1.0, time_shift=5.0, samp_freq=25.6, samples=256):
- "Show fft(rect(at)) = 1/abs(a) * sinc(f/a)"
- samp_freq = _numpy.float(samp_freq)
- a = _numpy.float(a)
-
- x = _numpy.zeros((samples,), dtype=_numpy.float)
- dt = 1.0/samp_freq
- for i in range(samples):
- t = i*dt
- x[i] = _gaussian(a, (t-time_shift))
- freq_axis, X = unitary_rfft(x, samp_freq)
- #_test_unitary_rfft_parsevals(x, samp_freq, freq_axis, X)
-
- # remove the phase due to our time shift
- j = _numpy.complex(0.0,1.0) # sqrt(-1)
- for i in range(len(freq_axis)):
- f = freq_axis[i]
- inverse_phase_shift = _numpy.exp(j*2.0*_numpy.pi*time_shift*f)
- X[i] *= inverse_phase_shift
-
- expected = _numpy.zeros((len(freq_axis),), dtype=_numpy.float)
- for i in range(len(freq_axis)):
- f = freq_axis[i]
- # see Wikipedia, or do the integral yourself.
- expected[i] = _numpy.sqrt(_numpy.pi/a) * _gaussian(
- 1.0/a, _numpy.pi*f)
-
- if TEST_PLOTS:
- figure = _pyplot.figure()
- time_axes = figure.add_subplot(2, 1, 1)
- time_axes.plot(_numpy.arange(0, dt*samples, dt), x)
- time_axes.set_title('time series')
- freq_axes = figure.add_subplot(2, 1, 2)
- freq_axes.plot(freq_axis, X.real, 'r.')
- freq_axes.plot(freq_axis, X.imag, 'g.')
- freq_axes.plot(freq_axis, expected, 'b-')
- freq_axes.set_title('freq series')
-
-
-def _test_unitary_rfft_gaussian_suite():
- print("Test unitary FFTs on variously shaped gaussian functions")
- _test_unitary_rfft_gaussian(a=0.5)
- _test_unitary_rfft_gaussian(a=2.0)
- _test_unitary_rfft_gaussian(a=0.7, samp_freq=50, samples=512)
- _test_unitary_rfft_gaussian(a=3.0, samp_freq=60, samples=1024)
-
-
def power_spectrum(data, freq=1.0):
- """Compute the power spectrum of DATA taken with a sampling frequency FREQ.
-
- DATA must be real (not complex).
- Returns a tuple of two arrays, (freq_axis, power), suitable for plotting.
- If the number of samples in data is not an integer power of two,
+ """Compute the power spectrum of the time series `data`.
+
+ Parameters
+ ----------
+ data : iterable
+ Real (not complex) data taken with a sampling frequency `freq`.
+ freq : real
+ Sampling frequency.
+
+ Returns
+ -------
+ freq_axis,power : numpy.ndarray
+ Arrays ready for plotting.
+
+ Notes
+ -----
+ If the number of samples in `data` is not an integer power of two,
the FFT ignores some of the later points.
+
+ See Also
+ --------
+ unitary_power_spectrum,avg_power_spectrum
"""
nsamps = floor_pow_of_two(len(data))
- freq_axis = _numpy.linspace(0, freq/2, nsamps/2+1)
+ freq_axis = _numpy.linspace(0, freq / 2, nsamps / 2 + 1)
# nsamps/2+1 b/c zero-freq and nyqist-freq are both fully real.
# >>> help(numpy.fft.fftpack.rfft) for Numpy's explaination.
# See Numerical Recipies for a details.
def unitary_power_spectrum(data, freq=1.0):
+ """Compute the unitary power spectrum of the time series `data`.
+
+ See Also
+ --------
+ power_spectrum,unitary_avg_power_spectrum
+ """
freq_axis,power = power_spectrum(data, freq)
# One sided power spectral density, so 2|H(f)|**2
# (see NR 2nd edition 12.0.14, p498)
return (freq_axis, power)
-def _test_unitary_power_spectrum_sin(sin_freq=10, samp_freq=512, samples=1024):
- x = _numpy.zeros((samples,), dtype=_numpy.float)
- samp_freq = _numpy.float(samp_freq)
- for i in range(samples):
- x[i] = _numpy.sin(2.0 * _numpy.pi * (i/samp_freq) * sin_freq)
- freq_axis, power = unitary_power_spectrum(x, samp_freq)
- imax = _numpy.argmax(power)
-
- expected = _numpy.zeros((len(freq_axis),), dtype=_numpy.float)
- df = samp_freq/_numpy.float(samples) # df = 1/T, where T = total_time
- i = int(sin_freq/df)
- # average power per unit time is
- # P = <x(t)**2>
- # average value of sin(t)**2 = 0.5 (b/c sin**2+cos**2 == 1)
- # so average value of (int sin(t)**2 dt) per unit time is 0.5
- # P = 0.5
- # we spread that power over a frequency bin of width df, sp
- # P(f0) = 0.5/df
- # where f0 is the sin's frequency
- #
- # or:
- # FFT of sin(2*pi*t*f0) gives
- # X(f) = 0.5 i (delta(f-f0) - delta(f-f0)),
- # (area under x(t) = 0, area under X(f) = 0)
- # so one sided power spectral density (PSD) per unit time is
- # P(f) = 2 |X(f)|**2 / T
- # = 2 * |0.5 delta(f-f0)|**2 / T
- # = 0.5 * |delta(f-f0)|**2 / T
- # but we're discrete and want the integral of the 'delta' to be 1,
- # so 'delta'*df = 1 --> 'delta' = 1/df, and
- # P(f) = 0.5 / (df**2 * T)
- # = 0.5 / df (T = 1/df)
- expected[i] = 0.5 / df
-
- print('The power should be a peak at {} Hz of {} ({}, {})'.format(
- sin_freq, expected[i], freq_axis[imax], power[imax]))
- Pexp = 0
- P = 0
- for i in range(len(freq_axis)):
- Pexp += expected[i] *df
- P += power[i] * df
- print('The total power should be {} ({})'.format(Pexp, P))
-
- if TEST_PLOTS:
- figure = _pyplot.figure()
- time_axes = figure.add_subplot(2, 1, 1)
- time_axes.plot(
- _numpy.arange(0, samples/samp_freq, 1.0/samp_freq), x, 'b-')
- time_axes.set_title('time series')
- freq_axes = figure.add_subplot(2, 1, 2)
- freq_axes.plot(freq_axis, power, 'r.')
- freq_axes.plot(freq_axis, expected, 'b-')
- freq_axes.set_title(
- '{} samples of sin at {} Hz'.format(samples, sin_freq))
-
-
-def _test_unitary_power_spectrum_sin_suite():
- print('Test unitary power spectrums on variously shaped sin functions')
- _test_unitary_power_spectrum_sin(sin_freq=5, samp_freq=512, samples=1024)
- _test_unitary_power_spectrum_sin(sin_freq=5, samp_freq=512, samples=2048)
- _test_unitary_power_spectrum_sin(sin_freq=5, samp_freq=512, samples=4098)
- _test_unitary_power_spectrum_sin(sin_freq=7, samp_freq=512, samples=1024)
- _test_unitary_power_spectrum_sin(sin_freq=5, samp_freq=1024, samples=2048)
- # with some irrational numbers, to check that I'm not getting lucky
- _test_unitary_power_spectrum_sin(
- sin_freq=_numpy.pi, samp_freq=100*_numpy.exp(1), samples=1024)
- # test with non-integer number of periods
- _test_unitary_power_spectrum_sin(sin_freq=5, samp_freq=512, samples=256)
-
-
-def _test_unitary_power_spectrum_delta(amp=1, samp_freq=1, samples=256):
- x = _numpy.zeros((samples,), dtype=_numpy.float)
- samp_freq = _numpy.float(samp_freq)
- x[0] = amp
- freq_axis, power = unitary_power_spectrum(x, samp_freq)
-
- # power = <x(t)**2> = (amp)**2 * dt/T
- # we spread that power over the entire freq_axis [0,fN], so
- # P(f) = (amp)**2 dt / (T fN)
- # where
- # dt = 1/samp_freq (sample period)
- # T = samples/samp_freq (total time of data aquisition)
- # fN = 0.5 samp_freq (Nyquist frequency)
- # so
- # P(f) = amp**2 / (samp_freq * samples/samp_freq * 0.5 samp_freq)
- # = 2 amp**2 / (samp_freq*samples)
- expected_amp = 2.0 * amp**2 / (samp_freq * samples)
- expected = _numpy.ones(
- (len(freq_axis),), dtype=_numpy.float) * expected_amp
-
- print('The power should be flat at y = {} ({})'.format(
- expected_amp, power[0]))
-
- if TEST_PLOTS:
- figure = _pyplot.figure()
- time_axes = figure.add_subplot(2, 1, 1)
- time_axes.plot(
- _numpy.arange(0, samples/samp_freq, 1.0/samp_freq), x, 'b-')
- time_axes.set_title('time series')
- freq_axes = figure.add_subplot(2, 1, 2)
- freq_axes.plot(freq_axis, power, 'r.')
- freq_axes.plot(freq_axis, expected, 'b-')
- freq_axes.set_title('{} samples of delta amp {}'.format(samples, amp))
-
-
-def _test_unitary_power_spectrum_delta_suite():
- print('Test unitary power spectrums on various delta functions')
- _test_unitary_power_spectrum_delta(amp=1, samp_freq=1.0, samples=1024)
- _test_unitary_power_spectrum_delta(amp=1, samp_freq=1.0, samples=2048)
- # expected = 2*computed
- _test_unitary_power_spectrum_delta(amp=1, samp_freq=0.5, samples=2048)
- # expected = 0.5*computed
- _test_unitary_power_spectrum_delta(amp=1, samp_freq=2.0, samples=2048)
- _test_unitary_power_spectrum_delta(amp=3, samp_freq=1.0, samples=1024)
- _test_unitary_power_spectrum_delta(
- amp=_numpy.pi, samp_freq=_numpy.exp(1), samples=1024)
-
-
-def _gaussian2(area, mean, std, t):
- "Integral over all time = area (i.e. normalized for area=1)"
- return area/(std*_numpy.sqrt(2.0*_numpy.pi)) * _numpy.exp(
- -0.5*((t-mean)/std)**2)
-
-
-def _test_unitary_power_spectrum_gaussian(
- area=2.5, mean=5, std=1, samp_freq=10.24 ,samples=512):
- x = _numpy.zeros((samples,), dtype=_numpy.float)
- mean = _numpy.float(mean)
- for i in range(samples):
- t = i/_numpy.float(samp_freq)
- x[i] = _gaussian2(area, mean, std, t)
- freq_axis, power = unitary_power_spectrum(x, samp_freq)
-
- # generate the predicted curve
- # by comparing our _gaussian2() form to _gaussian(),
- # we see that the Fourier transform of x(t) has parameters:
- # std' = 1/(2 pi std) (references declaring std' = 1/std are
- # converting to angular frequency,
- # not frequency like we are)
- # area' = area/[std sqrt(2*pi)] (plugging into FT of _gaussian() above)
- # mean' = 0 (changing the mean in the time-domain just
- # changes the phase in the freq-domain)
- # So our power spectral density per unit time is given by
- # P(f) = 2 |X(f)|**2 / T
- # Where
- # T = samples/samp_freq (total time of data aquisition)
- mean = 0.0
- area = area /(std*_numpy.sqrt(2.0*_numpy.pi))
- std = 1.0/(2.0*_numpy.pi*std)
- expected = _numpy.zeros((len(freq_axis),), dtype=_numpy.float)
- # 1/total_time ( = freq_axis[1]-freq_axis[0] = freq_axis[1])
- df = _numpy.float(samp_freq)/samples
- for i in range(len(freq_axis)):
- f = i*df
- gaus = _gaussian2(area, mean, std, f)
- expected[i] = 2.0 * gaus**2 * samp_freq/samples
- print(('The power should be a half-gaussian, '
- 'with a peak at 0 Hz with amplitude {} ({})').format(
- expected[0], power[0]))
-
- if TEST_PLOTS:
- figure = _pyplot.figure()
- time_axes = figure.add_subplot(2, 1, 1)
- time_axes.plot(
- _numpy.arange(0, samples/samp_freq, 1.0/samp_freq), x, 'b-')
- time_axes.set_title('time series')
- freq_axes = figure.add_subplot(2, 1, 2)
- freq_axes.plot(freq_axis, power, 'r.')
- freq_axes.plot(freq_axis, expected, 'b-')
- freq_axes.set_title('freq series')
-
-
-def _test_unitary_power_spectrum_gaussian_suite():
- print('Test unitary power spectrums on various gaussian functions')
- _test_unitary_power_spectrum_gaussian(
- area=1, std=1, samp_freq=10.0, samples=1024)
- _test_unitary_power_spectrum_gaussian(
- area=1, std=2, samp_freq=10.0, samples=1024)
- _test_unitary_power_spectrum_gaussian(
- area=1, std=1, samp_freq=10.0, samples=2048)
- _test_unitary_power_spectrum_gaussian(
- area=1, std=1, samp_freq=20.0, samples=2048)
- _test_unitary_power_spectrum_gaussian(
- area=3, std=1, samp_freq=10.0, samples=1024)
- _test_unitary_power_spectrum_gaussian(
- area=_numpy.pi, std=_numpy.sqrt(2), samp_freq=_numpy.exp(1),
- samples=1024)
+def window_hann(length):
+ r"""Returns a Hann window array with length entries
+ Notes
+ -----
+ The Hann window with length :math:`L` is defined as
-def window_hann(length):
- "Returns a Hann window array with length entries"
+ .. math:: w_i = \frac{1}{2} (1-\cos(2\pi i/L))
+ """
win = _numpy.zeros((length,), dtype=_numpy.float)
for i in range(length):
- win[i] = 0.5*(1.0-_numpy.cos(2.0*_numpy.pi*_numpy.float(i)/(length)))
+ win[i] = 0.5 * (
+ 1.0 - _numpy.cos(2.0 * _numpy.pi * _numpy.float(i) / (length)))
# avg value of cos over a period is 0
# so average height of Hann window is 0.5
return win
def avg_power_spectrum(data, freq=1.0, chunk_size=2048,
overlap=True, window=window_hann):
- """Compute the avgerage power spectrum of DATA.
-
- Taken with a sampling frequency FREQ.
-
- DATA must be real (not complex) by breaking DATA into chunks.
- The chunks may or may not be overlapping (by setting OVERLAP).
- The chunks are windowed by dotting with WINDOW(CHUNK_SIZE), FFTed,
- and the resulting spectra are averaged together.
- See NR 13.4 for rational.
-
- Returns a tuple of two arrays, (freq_axis, power), suitable for plotting.
- CHUNK_SIZE should really be a power of 2.
- If the number of samples in DATA is not an integer power of CHUNK_SIZE,
- the FFT ignores some of the later points.
+ """Compute the avgerage power spectrum of `data`.
+
+ Parameters
+ ----------
+ data : iterable
+ Real (not complex) data taken with a sampling frequency `freq`.
+ freq : real
+ Sampling frequency.
+ chunk_size : int
+ Number of samples per chunk. Use a power of two.
+ overlap: {True,False}
+ If `True`, each chunk overlaps the previous chunk by half its
+ length. Otherwise, the chunks are end-to-end, and not
+ overlapping.
+ window: iterable
+ Weights used to "smooth" the chunks, there is a whole science
+ behind windowing, but if you're not trying to squeeze every
+ drop of information out of your data, you'll be OK with the
+ default Hann window.
+
+ Returns
+ -------
+ freq_axis,power : numpy.ndarray
+ Arrays ready for plotting.
+
+ Notes
+ -----
+ The average power spectrum is computed by breaking `data` into
+ chunks of length `chunk_size`. These chunks are transformed
+ individually into frequency space and then averaged together.
+
+ See Numerical Recipes 2 section 13.4 for a good introduction to
+ the theory.
+
+ If the number of samples in `data` is not a multiple of
+ `chunk_size`, we ignore the extra points.
"""
if chunk_size != floor_pow_of_two(chunk_size):
raise ValueError(
'chunk_size {} should be a power of 2'.format(chunk_size))
- nchunks = len(data)/chunk_size # integer division = implicit floor
+ nchunks = len(data) // chunk_size # integer division = implicit floor
if overlap:
- chunk_step = chunk_size/2
+ chunk_step = chunk_size / 2
else:
chunk_step = chunk_size
win = window(chunk_size) # generate a window of the appropriate size
- freq_axis = _numpy.linspace(0, freq/2, chunk_size/2+1)
+ freq_axis = _numpy.linspace(0, freq / 2, chunk_size / 2 + 1)
# nsamps/2+1 b/c zero-freq and nyqist-freq are both fully real.
# >>> help(numpy.fft.fftpack.rfft) for Numpy's explaination.
# See Numerical Recipies for a details.
- power = _numpy.zeros((chunk_size/2+1,), dtype=_numpy.float)
+ power = _numpy.zeros((chunk_size / 2 + 1, ), dtype=_numpy.float)
for i in range(nchunks):
- starti = i*chunk_step
- stopi = starti+chunk_size
- fft_chunk = _numpy.fft.rfft(data[starti:stopi]*win)
+ starti = i * chunk_step
+ stopi = starti + chunk_size
+ fft_chunk = _numpy.fft.rfft(data[starti:stopi] * win)
p_chunk = (fft_chunk * fft_chunk.conj()).real
power += p_chunk.astype(_numpy.float)
power /= _numpy.float(nchunks)
def unitary_avg_power_spectrum(data, freq=1.0, chunk_size=2048,
overlap=True, window=window_hann):
- """Compute the average power spectrum, preserving normalization
+ """Compute the unitary average power spectrum of `data`.
+
+ See Also
+ --------
+ avg_power_spectrum,unitary_power_spectrum
"""
freq_axis,power = avg_power_spectrum(
data, freq, chunk_size, overlap, window)
# 2.0 / (freq * chunk_size) |rfft()|**2 --> unitary_power_spectrum
# see unitary_power_spectrum()
- power *= 2.0 / (freq*_numpy.float(chunk_size)) * 8/3
+ power *= 2.0 / (freq * _numpy.float(chunk_size)) * 8.0 / 3
# * 8/3 to remove power from windowing
# <[x(t)*w(t)]**2> = <x(t)**2 * w(t)**2> ~= <x(t)**2> * <w(t)**2>
# where the ~= is because the frequency of x(t) >> the frequency of w(t).
return (freq_axis, power)
-def _test_unitary_avg_power_spectrum_sin(
- sin_freq=10, samp_freq=512, samples=1024, chunk_size=512, overlap=True,
- window=window_hann):
- x = _numpy.zeros((samples,), dtype=_numpy.float)
- samp_freq = _numpy.float(samp_freq)
- for i in range(samples):
- x[i] = _numpy.sin(2.0 * _numpy.pi * (i/samp_freq) * sin_freq)
- freq_axis, power = unitary_avg_power_spectrum(
- x, samp_freq, chunk_size, overlap, window)
- imax = _numpy.argmax(power)
-
- expected = _numpy.zeros((len(freq_axis),), dtype=_numpy.float)
- df = samp_freq/_numpy.float(chunk_size) # df = 1/T, where T = total_time
- i = int(sin_freq/df)
- expected[i] = 0.5 / df # see _test_unitary_power_spectrum_sin()
-
- print('The power should peak at {} Hz of {} ({}, {})'.format(
- sin_freq, expected[i], freq_axis[imax], power[imax]))
- Pexp = 0
- P = 0
- for i in range(len(freq_axis)):
- Pexp += expected[i] * df
- P += power[i] * df
- print('The total power should be {} ({})'.format(Pexp, P))
-
- if TEST_PLOTS:
- figure = _pyplot.figure()
- time_axes = figure.add_subplot(2, 1, 1)
- time_axes.plot(
- _numpy.arange(0, samples/samp_freq, 1.0/samp_freq), x, 'b-')
- time_axes.set_title('time series')
- freq_axes = figure.add_subplot(2, 1, 2)
- freq_axes.plot(freq_axis, power, 'r.')
- freq_axes.plot(freq_axis, expected, 'b-')
- freq_axes.set_title(
- '{} samples of sin at {} Hz'.format(samples, sin_freq))
-
-
-def _test_unitary_avg_power_spectrum_sin_suite():
- print('Test unitary avg power spectrums on variously shaped sin functions')
- _test_unitary_avg_power_spectrum_sin(
- sin_freq=5, samp_freq=512, samples=1024)
- _test_unitary_avg_power_spectrum_sin(
- sin_freq=5, samp_freq=512, samples=2048)
- _test_unitary_avg_power_spectrum_sin(
- sin_freq=5, samp_freq=512, samples=4098)
- _test_unitary_avg_power_spectrum_sin(
- sin_freq=17, samp_freq=512, samples=1024)
- _test_unitary_avg_power_spectrum_sin(
- sin_freq=5, samp_freq=1024, samples=2048)
- # test long wavelenth sin, so be closer to window frequency
- _test_unitary_avg_power_spectrum_sin(
- sin_freq=1, samp_freq=1024, samples=2048)
- # finally, with some irrational numbers, to check that I'm not
- # getting lucky
- _test_unitary_avg_power_spectrum_sin(
- sin_freq=_numpy.pi, samp_freq=100*_numpy.exp(1), samples=1024)
-
-
-def test():
- _test_rfft_suite()
- _test_unitary_rfft_parsevals_suite()
- _test_unitary_rfft_rect_suite()
- _test_unitary_rfft_gaussian_suite()
- _test_unitary_power_spectrum_sin_suite()
- _test_unitary_power_spectrum_delta_suite()
- _test_unitary_power_spectrum_gaussian_suite()
- _test_unitary_avg_power_spectrum_sin_suite()
-
-
-if __name__ == '__main__':
- from optparse import OptionParser
-
- p = OptionParser('%prog [options]', epilog='Run FFT-tools unit tests.')
- p.add_option('-p', '--plot', dest='plot', action='store_true',
- help='Display time- and freq-space plots of test transforms.')
-
- options,args = p.parse_args()
-
- if options.plot:
- import matplotlib.pyplot as _pyplot
- TEST_PLOTS = True
- test()
- if options.plot:
- _pyplot.show()
+class TestRFFT (_unittest.TestCase):
+ r"""Ensure Numpy's FFT algorithm acts as expected.
+
+ Notes
+ -----
+ The expected return values are [#dft]_:
+
+ .. math:: X_k = \sum_{m=0}^{n-1} x_m \exp^{-2\pi imk/n}
+
+ .. [#dft] See the *Background information* section of
+ :py:mod:`numpy.fft`.
+ """
+ def run_rfft(self, xs, Xs):
+ i = _numpy.complex(0, 1)
+ n = len(xs)
+ Xa = []
+ for k in range(n):
+ Xa.append(sum([x * _numpy.exp(-2 * _numpy.pi * i * m * k / n)
+ for x,m in zip(xs, range(n))]))
+ if k < len(Xs):
+ self.assertAlmostEqual(
+ (Xs[k] - Xa[k]) / _numpy.abs(Xa[k]), 0, 6,
+ ('rfft mismatch on element {}: {} != {}, '
+ 'relative error {}').format(
+ k, Xs[k], Xa[k], (Xs[k] - Xa[k]) / _numpy.abs(Xa[k])))
+ # Which should satisfy the discrete form of Parseval's theorem
+ # n-1 n-1
+ # SUM |x_m|^2 = 1/n SUM |X_k|^2.
+ # m=0 k=0
+ timeSum = sum([_numpy.abs(x)**2 for x in xs])
+ freqSum = sum([_numpy.abs(X)**2 for X in Xa])
+ self.assertAlmostEqual(
+ _numpy.abs(freqSum / _numpy.float(n) - timeSum) / timeSum, 0, 6,
+ "Mismatch on Parseval's, {} != 1/{} * {}".format(
+ timeSum, n, freqSum))
+
+ def test_rfft(self):
+ "Test NumPy's builtin :py:func:`numpy.fft.rfft`"
+ xs = [1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1]
+ self.run_rfft(xs, _numpy.fft.rfft(xs))
+
+
+class TestUnitaryRFFT (_unittest.TestCase):
+ """Verify :py:func:`unitary_rfft`.
+ """
+ def run_parsevals(self, xs, freq, freqs, Xs):
+ """Check the discretized integral form of Parseval's theorem
+
+ Notes
+ -----
+ Which is:
+
+ .. math:: \sum_{m=0}^{n-1} |x_m|^2 dt = \sum_{k=0}^{n-1} |X_k|^2 df
+ """
+ dt = 1.0 / freq
+ df = freqs[1] - freqs[0]
+ self.assertAlmostEqual(
+ (df - 1 / (len(xs) * dt)) / df, 0, 6,
+ 'Mismatch in spacing, {} != 1/({}*{})'.format(df, len(xs), dt))
+ Xa = list(Xs)
+ for k in range(len(Xs) - 1, 1, -1):
+ Xa.append(Xa[k])
+ self.assertEqual(len(xs), len(Xa))
+ lhs = sum([_numpy.abs(x)**2 for x in xs]) * dt
+ rhs = sum([_numpy.abs(X)**2 for X in Xa]) * df
+ self.assertAlmostEqual(
+ _numpy.abs(lhs - rhs) / lhs, 0, 3,
+ "Mismatch on Parseval's, {} != {}".format(lhs, rhs))
+
+ def test_parsevals(self):
+ "Test unitary rfft on Parseval's theorem"
+ xs = [1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1, 2, 3, 1]
+ dt = _numpy.pi
+ freqs,Xs = unitary_rfft(xs, 1.0 / dt)
+ self.run_parsevals(xs, 1.0 / dt, freqs, Xs)
+
+ def rect(self, t):
+ r"""Rectangle function.
+
+ Notes
+ -----
+
+ .. math::
+
+ \rect(t) = \begin{cases}
+ 1& \text{if $|t| < 0.5$}, \\
+ 0& \text{if $|t| \ge 0.5$}.
+ \end{cases}
+ """
+ if _numpy.abs(t) < 0.5:
+ return 1
+ else:
+ return 0
+
+ def run_rect(self, a=1.0, time_shift=5.0, samp_freq=25.6, samples=256):
+ r"""Test :py:func:`unitary_rfft` on known function :py:meth:`rect`.
+
+ Notes
+ -----
+ Analytic result:
+
+ .. math:: \rfft(\rect(at)) = 1/|a|\cdot\sinc(f/a)
+ """
+ samp_freq = _numpy.float(samp_freq)
+ a = _numpy.float(a)
+
+ x = _numpy.zeros((samples,), dtype=_numpy.float)
+ dt = 1.0 / samp_freq
+ for i in range(samples):
+ t = i * dt
+ x[i] = self.rect(a * (t - time_shift))
+ freq_axis,X = unitary_rfft(x, samp_freq)
+
+ # remove the phase due to our time shift
+ j = _numpy.complex(0.0, 1.0) # sqrt(-1)
+ for i in range(len(freq_axis)):
+ f = freq_axis[i]
+ inverse_phase_shift = _numpy.exp(
+ j * 2.0 * _numpy.pi * time_shift * f)
+ X[i] *= inverse_phase_shift
+
+ expected = _numpy.zeros((len(freq_axis),), dtype=_numpy.float)
+ # normalized sinc(x) = sin(pi x)/(pi x)
+ # so sinc(0.5) = sin(pi/2)/(pi/2) = 2/pi
+ self.assertEqual(_numpy.sinc(0.5), 2.0 / _numpy.pi)
+ for i in range(len(freq_axis)):
+ f = freq_axis[i]
+ expected[i] = 1.0 / _numpy.abs(a) * _numpy.sinc(f / a)
+
+ if TEST_PLOTS:
+ if _pyplot is None:
+ raise _pyplot_import_error
+ figure = _pyplot.figure()
+ time_axes = figure.add_subplot(2, 1, 1)
+ time_axes.plot(_numpy.arange(0, dt * samples, dt), x)
+ time_axes.set_title('time series')
+ freq_axes = figure.add_subplot(2, 1, 2)
+ freq_axes.plot(freq_axis, X.real, 'r.')
+ freq_axes.plot(freq_axis, X.imag, 'g.')
+ freq_axes.plot(freq_axis, expected, 'b-')
+ freq_axes.set_title('freq series')
+
+ def test_rect(self):
+ "Test unitary FFTs on variously shaped rectangular functions."
+ self.run_rect(a=0.5)
+ self.run_rect(a=2.0)
+ self.run_rect(a=0.7, samp_freq=50, samples=512)
+ self.run_rect(a=3.0, samp_freq=60, samples=1024)
+
+ def gaussian(self, a, t):
+ r"""Gaussian function.
+
+ Notes
+ -----
+
+ .. math:: \gaussian(a,t) = \exp^{-at^2}
+ """
+ return _numpy.exp(-a * t**2)
+
+ def run_gaussian(self, a=1.0, time_shift=5.0, samp_freq=25.6, samples=256):
+ r"""Test :py:func:`unitary_rttf` on known function :py:meth:`gaussian`.
+
+ Notes
+ -----
+ Analytic result:
+
+ .. math::
+
+ \rfft(\gaussian(a,t))
+ = \sqrt{\pi/a} \cdot \gaussian(1/a,\pi f)
+ """
+ samp_freq = _numpy.float(samp_freq)
+ a = _numpy.float(a)
+
+ x = _numpy.zeros((samples,), dtype=_numpy.float)
+ dt = 1.0 / samp_freq
+ for i in range(samples):
+ t = i * dt
+ x[i] = self.gaussian(a, (t - time_shift))
+ freq_axis,X = unitary_rfft(x, samp_freq)
+
+ # remove the phase due to our time shift
+ j = _numpy.complex(0.0, 1.0) # sqrt(-1)
+ for i in range(len(freq_axis)):
+ f = freq_axis[i]
+ inverse_phase_shift = _numpy.exp(
+ j * 2.0 * _numpy.pi * time_shift * f)
+ X[i] *= inverse_phase_shift
+
+ expected = _numpy.zeros((len(freq_axis),), dtype=_numpy.float)
+ for i in range(len(freq_axis)):
+ f = freq_axis[i]
+ # see Wikipedia, or do the integral yourself.
+ expected[i] = _numpy.sqrt(_numpy.pi / a) * self.gaussian(
+ 1.0 / a, _numpy.pi * f)
+
+ if TEST_PLOTS:
+ if _pyplot is None:
+ raise _pyplot_import_error
+ figure = _pyplot.figure()
+ time_axes = figure.add_subplot(2, 1, 1)
+ time_axes.plot(_numpy.arange(0, dt * samples, dt), x)
+ time_axes.set_title('time series')
+ freq_axes = figure.add_subplot(2, 1, 2)
+ freq_axes.plot(freq_axis, X.real, 'r.')
+ freq_axes.plot(freq_axis, X.imag, 'g.')
+ freq_axes.plot(freq_axis, expected, 'b-')
+ freq_axes.set_title('freq series')
+
+ def test_gaussian(self):
+ "Test unitary FFTs on variously shaped gaussian functions."
+ self.run_gaussian(a=0.5)
+ self.run_gaussian(a=2.0)
+ self.run_gaussian(a=0.7, samp_freq=50, samples=512)
+ self.run_gaussian(a=3.0, samp_freq=60, samples=1024)
+
+
+class TestUnitaryPowerSpectrum (_unittest.TestCase):
+ def run_sin(self, sin_freq=10, samp_freq=512, samples=1024):
+ x = _numpy.zeros((samples,), dtype=_numpy.float)
+ samp_freq = _numpy.float(samp_freq)
+ for i in range(samples):
+ x[i] = _numpy.sin(2.0 * _numpy.pi * (i / samp_freq) * sin_freq)
+ freq_axis,power = unitary_power_spectrum(x, samp_freq)
+ imax = _numpy.argmax(power)
+
+ expected = _numpy.zeros((len(freq_axis),), dtype=_numpy.float)
+ # df = 1/T, where T = total_time
+ df = samp_freq / _numpy.float(samples)
+ i = int(sin_freq / df)
+ # average power per unit time is
+ # P = <x(t)**2>
+ # average value of sin(t)**2 = 0.5 (b/c sin**2+cos**2 == 1)
+ # so average value of (int sin(t)**2 dt) per unit time is 0.5
+ # P = 0.5
+ # we spread that power over a frequency bin of width df, sp
+ # P(f0) = 0.5/df
+ # where f0 is the sin's frequency
+ #
+ # or:
+ # FFT of sin(2*pi*t*f0) gives
+ # X(f) = 0.5 i (delta(f-f0) - delta(f-f0)),
+ # (area under x(t) = 0, area under X(f) = 0)
+ # so one sided power spectral density (PSD) per unit time is
+ # P(f) = 2 |X(f)|**2 / T
+ # = 2 * |0.5 delta(f-f0)|**2 / T
+ # = 0.5 * |delta(f-f0)|**2 / T
+ # but we're discrete and want the integral of the 'delta' to be 1,
+ # so 'delta'*df = 1 --> 'delta' = 1/df, and
+ # P(f) = 0.5 / (df**2 * T)
+ # = 0.5 / df (T = 1/df)
+ expected[i] = 0.5 / df
+
+ LOG.debug('The power should be a peak at {} Hz of {} ({}, {})'.format(
+ sin_freq, expected[i], freq_axis[imax], power[imax]))
+ Pexp = P = 0
+ for i in range(len(freq_axis)):
+ Pexp += expected[i] * df
+ P += power[i] * df
+ self.assertAlmostEqual(
+ _numpy.abs((P - Pexp) / Pexp), 0, 1,
+ 'The total power should be {} ({})'.format(Pexp, P))
+
+ if TEST_PLOTS:
+ if _pyplot is None:
+ raise _pyplot_import_error
+ figure = _pyplot.figure()
+ time_axes = figure.add_subplot(2, 1, 1)
+ time_axes.plot(
+ _numpy.arange(0, samples / samp_freq, 1.0 / samp_freq), x, 'b-')
+ time_axes.set_title('time series')
+ freq_axes = figure.add_subplot(2, 1, 2)
+ freq_axes.plot(freq_axis, power, 'r.')
+ freq_axes.plot(freq_axis, expected, 'b-')
+ freq_axes.set_title(
+ '{} samples of sin at {} Hz'.format(samples, sin_freq))
+
+ def test_sin(self):
+ "Test unitary power spectrums on variously shaped sin functions"
+ self.run_sin(sin_freq=5, samp_freq=512, samples=1024)
+ self.run_sin(sin_freq=5, samp_freq=512, samples=2048)
+ self.run_sin(sin_freq=5, samp_freq=512, samples=4098)
+ self.run_sin(sin_freq=7, samp_freq=512, samples=1024)
+ self.run_sin(sin_freq=5, samp_freq=1024, samples=2048)
+ # finally, with some irrational numbers, to check that I'm not
+ # getting lucky
+ self.run_sin(
+ sin_freq=_numpy.pi, samp_freq=100 * _numpy.exp(1), samples=1024)
+ # test with non-integer number of periods
+ self.run_sin(sin_freq=5, samp_freq=512, samples=256)
+
+ def run_delta(self, amp=1, samp_freq=1, samples=256):
+ """TODO
+ """
+ x = _numpy.zeros((samples,), dtype=_numpy.float)
+ samp_freq = _numpy.float(samp_freq)
+ x[0] = amp
+ freq_axis,power = unitary_power_spectrum(x, samp_freq)
+
+ # power = <x(t)**2> = (amp)**2 * dt/T
+ # we spread that power over the entire freq_axis [0,fN], so
+ # P(f) = (amp)**2 dt / (T fN)
+ # where
+ # dt = 1/samp_freq (sample period)
+ # T = samples/samp_freq (total time of data aquisition)
+ # fN = 0.5 samp_freq (Nyquist frequency)
+ # so
+ # P(f) = amp**2 / (samp_freq * samples/samp_freq * 0.5 samp_freq)
+ # = 2 amp**2 / (samp_freq*samples)
+ expected_amp = 2.0 * amp**2 / (samp_freq * samples)
+ expected = _numpy.ones(
+ (len(freq_axis),), dtype=_numpy.float) * expected_amp
+
+ self.assertAlmostEqual(
+ expected_amp, power[0], 4,
+ 'The power should be flat at y = {} ({})'.format(
+ expected_amp, power[0]))
+
+ if TEST_PLOTS:
+ if _pyplot is None:
+ raise _pyplot_import_error
+ figure = _pyplot.figure()
+ time_axes = figure.add_subplot(2, 1, 1)
+ time_axes.plot(
+ _numpy.arange(0, samples / samp_freq, 1.0 / samp_freq), x, 'b-')
+ time_axes.set_title('time series')
+ freq_axes = figure.add_subplot(2, 1, 2)
+ freq_axes.plot(freq_axis, power, 'r.')
+ freq_axes.plot(freq_axis, expected, 'b-')
+ freq_axes.set_title('{} samples of delta amp {}'.format(samples, amp))
+
+ def test_delta(self):
+ "Test unitary power spectrums on various delta functions"
+ self.run_delta(amp=1, samp_freq=1.0, samples=1024)
+ self.run_delta(amp=1, samp_freq=1.0, samples=2048)
+ # expected = 2*computed
+ self.run_delta(amp=1, samp_freq=0.5, samples=2048)
+ # expected = 0.5*computed
+ self.run_delta(amp=1, samp_freq=2.0, samples=2048)
+ self.run_delta(amp=3, samp_freq=1.0, samples=1024)
+ self.run_delta(amp=_numpy.pi, samp_freq=_numpy.exp(1), samples=1024)
+
+ def gaussian(self, area, mean, std, t):
+ "Integral over all time = area (i.e. normalized for area=1)"
+ return area / (std * _numpy.sqrt(2.0 * _numpy.pi)) * _numpy.exp(
+ -0.5 * ((t-mean)/std)**2)
+
+ def run_gaussian(self, area=2.5, mean=5, std=1, samp_freq=10.24,
+ samples=512):
+ """TODO.
+ """
+ x = _numpy.zeros((samples,), dtype=_numpy.float)
+ mean = _numpy.float(mean)
+ for i in range(samples):
+ t = i / _numpy.float(samp_freq)
+ x[i] = self.gaussian(area, mean, std, t)
+ freq_axis,power = unitary_power_spectrum(x, samp_freq)
+
+ # generate the predicted curve by comparing our
+ # TestUnitaryPowerSpectrum.gaussian() form to
+ # TestUnitaryRFFT.gaussian(),
+ # we see that the Fourier transform of x(t) has parameters:
+ # std' = 1/(2 pi std) (references declaring std' = 1/std are
+ # converting to angular frequency,
+ # not frequency like we are)
+ # area' = area/[std sqrt(2*pi)] (plugging into FT of
+ # TestUnitaryRFFT.gaussian() above)
+ # mean' = 0 (changing the mean in the time-domain just
+ # changes the phase in the freq-domain)
+ # So our power spectral density per unit time is given by
+ # P(f) = 2 |X(f)|**2 / T
+ # Where
+ # T = samples/samp_freq (total time of data aquisition)
+ mean = 0.0
+ area = area / (std * _numpy.sqrt(2.0 * _numpy.pi))
+ std = 1.0 / (2.0 * _numpy.pi * std)
+ expected = _numpy.zeros((len(freq_axis),), dtype=_numpy.float)
+ # 1/total_time ( = freq_axis[1]-freq_axis[0] = freq_axis[1])
+ df = _numpy.float(samp_freq) / samples
+ for i in range(len(freq_axis)):
+ f = i * df
+ gaus = self.gaussian(area, mean, std, f)
+ expected[i] = 2.0 * gaus**2 * samp_freq / samples
+ self.assertAlmostEqual(
+ expected[0], power[0], 3,
+ ('The power should be a half-gaussian, '
+ 'with a peak at 0 Hz with amplitude {} ({})').format(
+ expected[0], power[0]))
+
+ if TEST_PLOTS:
+ if _pyplot is None:
+ raise _pyplot_import_error
+ figure = _pyplot.figure()
+ time_axes = figure.add_subplot(2, 1, 1)
+ time_axes.plot(
+ _numpy.arange(0, samples / samp_freq, 1.0 / samp_freq),
+ x, 'b-')
+ time_axes.set_title('time series')
+ freq_axes = figure.add_subplot(2, 1, 2)
+ freq_axes.plot(freq_axis, power, 'r.')
+ freq_axes.plot(freq_axis, expected, 'b-')
+ freq_axes.set_title('freq series')
+
+ def test_gaussian(self):
+ "Test unitary power spectrums on various gaussian functions"
+ for area in [1, _numpy.pi]:
+ for std in [1, _numpy.sqrt(2)]:
+ for samp_freq in [10.0, _numpy.exp(1)]:
+ for samples in [1024, 2048]:
+ self.run_gaussian(
+ area=area, std=std, samp_freq=samp_freq,
+ samples=samples)
+
+
+class TestUnitaryAvgPowerSpectrum (_unittest.TestCase):
+ def run_sin(self, sin_freq=10, samp_freq=512, samples=1024, chunk_size=512,
+ overlap=True, window=window_hann, places=3):
+ """TODO
+ """
+ x = _numpy.zeros((samples,), dtype=_numpy.float)
+ samp_freq = _numpy.float(samp_freq)
+ for i in range(samples):
+ x[i] = _numpy.sin(2.0 * _numpy.pi * (i / samp_freq) * sin_freq)
+ freq_axis,power = unitary_avg_power_spectrum(
+ x, samp_freq, chunk_size, overlap, window)
+ imax = _numpy.argmax(power)
+
+ expected = _numpy.zeros((len(freq_axis),), dtype=_numpy.float)
+ # df = 1/T, where T = total_time
+ df = samp_freq / _numpy.float(chunk_size)
+ i = int(sin_freq / df)
+ # see TestUnitaryPowerSpectrum.run_unitary_power_spectrum_sin()
+ expected[i] = 0.5 / df
+
+ LOG.debug('The power should peak at {} Hz of {} ({}, {})'.format(
+ sin_freq, expected[i], freq_axis[imax], power[imax]))
+ Pexp = P = 0
+ for i in range(len(freq_axis)):
+ Pexp += expected[i] * df
+ P += power[i] * df
+ self.assertAlmostEqual(
+ Pexp, P, places,
+ 'The total power should be {} ({})'.format(Pexp, P))
+
+ if TEST_PLOTS:
+ if _pyplot is None:
+ raise _pyplot_import_error
+ figure = _pyplot.figure()
+ time_axes = figure.add_subplot(2, 1, 1)
+ time_axes.plot(
+ _numpy.arange(0, samples / samp_freq, 1.0 / samp_freq),
+ x, 'b-')
+ time_axes.set_title('time series')
+ freq_axes = figure.add_subplot(2, 1, 2)
+ freq_axes.plot(freq_axis, power, 'r.')
+ freq_axes.plot(freq_axis, expected, 'b-')
+ freq_axes.set_title(
+ '{} samples of sin at {} Hz'.format(samples, sin_freq))
+
+ def test_sin(self):
+ "Test unitary avg power spectrums on variously shaped sin functions."
+ self.run_sin(sin_freq=5, samp_freq=512, samples=1024)
+ self.run_sin(sin_freq=5, samp_freq=512, samples=2048)
+ self.run_sin(sin_freq=5, samp_freq=512, samples=4098)
+ self.run_sin(sin_freq=17, samp_freq=512, samples=1024)
+ self.run_sin(sin_freq=5, samp_freq=1024, samples=2048)
+ # test long wavelenth sin, so be closer to window frequency
+ self.run_sin(sin_freq=1, samp_freq=1024, samples=2048, places=0)
+ # finally, with some irrational numbers, to check that I'm not
+ # getting lucky
+ self.run_sin(
+ sin_freq=_numpy.pi, samp_freq=100 * _numpy.exp(1), samples=1024)
projects / FFT-tools.git / blobdiff