-#!/usr/bin/python
+# Copyright (C) 2008-2012 W. Trevor King
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 3 of the License, or
+# (at your option) any later version.
#
-# Wrap Numpy's fft module to produce 1D unitary transforms and power spectra.
+# This program is distributed in the hope that it will be useful,
+# but WITHOUT ANY WARRANTY; without even the implied warranty of
+# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+# GNU General Public License for more details.
#
-# Copyright (C) 2008 W. Trevor King
-# All rights reserved.
-#
-# Redistribution and use in source and binary forms, with or without
-# modification, are permitted provided that the following conditions
-# are met:
-#
-# * Redistributions of source code must retain the above copyright
-# notice, this list of conditions and the following disclaimer.
-#
-# * Redistributions in binary form must reproduce the above copyright
-# notice, this list of conditions and the following disclaimer in
-# the documentation and/or other materials provided with the
-# distribution.
-#
-# * Neither the name of the copyright holders nor the names of the
-# contributors may be used to endorse or promote products derived
-# from this software without specific prior written permission.
-#
-# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
-# "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
-# LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
-# FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
-# COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
-# INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
-# BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES;
-# LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER
-# CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
-# LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
-# ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
-# POSSIBILITY OF SUCH DAMAGE.
+# You should have received a copy of the GNU General Public License
+# along with this program. If not, see <http://www.gnu.org/licenses/>.
-"""
-Define some FFT wrappers to reduce clutter.
-Provides a unitary discrete FFT and a windowed version.
-Based on :func:`numpy.fft.rfft`.
+"""Wrap Numpy's :py:mod:`~numpy.fft` module to reduce clutter.
+
+Provides a unitary discrete FFT and a windowed version based on
+:py:func:`numpy.fft.rfft`.
Main entry functions:
-* :func:`unitary_rfft`
-* :func:`power_spectrum`
-* :func:`unitary_power_spectrum`
-* :func:`avg_power_spectrum`
-* :func:`unitary_avg_power_spectrum`
+* :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 unittest
+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
+
-from numpy import log2, floor, round, ceil, abs, pi, exp, cos, sin, sqrt, \
- sinc, arctan2, array, ones, arange, linspace, zeros, \
- uint16, float, concatenate, fromfile, argmax, complex
-from numpy.fft import rfft
+__version__ = '0.5'
+LOG = _logging.getLogger('FFT-tools')
+LOG.addHandler(_logging.StreamHandler())
+LOG.setLevel(_logging.ERROR)
+
+
+# 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.
>>> floor_pow_of_two(15)
8
"""
- lnum = log2(num)
+ lnum = _numpy.log2(num)
if int(lnum) != lnum:
- num = 2**floor(lnum)
+ num = 2**_numpy.floor(lnum)
return int(num)
+
def round_pow_of_two(num):
"""Round `num` to the closest exact a power of two on a log scale.
>>> round_pow_of_two(15)
16
"""
- lnum = log2(num)
+ lnum = _numpy.log2(num)
if int(lnum) != lnum:
- num = 2**round(lnum)
+ num = 2**_numpy.round(lnum)
return int(num)
+
def ceil_pow_of_two(num):
"""Round `num` up to the closest exact a power of two.
>>> ceil_pow_of_two(15)
16
"""
- lnum = log2(num)
+ lnum = _numpy.log2(num)
if int(lnum) != lnum:
- num = 2**ceil(lnum)
+ num = 2**_numpy.ceil(lnum)
return int(num)
+
def unitary_rfft(data, freq=1.0):
"""Compute the unitary Fourier transform of real data.
nsamps = floor_pow_of_two(len(data))
# Which should satisfy the discrete form of Parseval's theorem
# n-1 n-1
- # SUM |x_m|^2 = 1/n SUM |X_k|^2.
+ # SUM |x_m|^2 = 1/n SUM |X_k|^2.
# m=0 k=0
# However, we want our FFT to satisfy the continuous Parseval eqn
# int_{-infty}^{infty} |x(t)|^2 dt = int_{-infty}^{infty} |X(f)|^2 df
# so we can convert the Numpy transformed data to match our unitary
# continuous transformed data with (also NR 12.1.8)
# X'_k = dtX = X / <sampling freq>
- trans = rfft(data[0:nsamps]) / float(freq)
- freq_axis = linspace(0, freq/2, nsamps/2+1)
+ trans = _numpy.fft.rfft(data[0:nsamps]) / _numpy.float(freq)
+ freq_axis = _numpy.linspace(0, freq / 2, nsamps / 2 + 1)
return (freq_axis, trans)
+
def power_spectrum(data, freq=1.0):
"""Compute the power spectrum of the time series `data`.
unitary_power_spectrum,avg_power_spectrum
"""
nsamps = floor_pow_of_two(len(data))
-
- freq_axis = 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.
- trans = rfft(data[0:nsamps])
- power = trans * trans.conj() # We want the square of the amplitude.
+ trans = _numpy.fft.rfft(data[0:nsamps])
+ power = (trans * trans.conj()).real # we want the square of the amplitude
return (freq_axis, power)
+
def unitary_power_spectrum(data, freq=1.0):
"""Compute the unitary power spectrum of the time series `data`.
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)
+ # One sided power spectral density, so 2|H(f)|**2
+ # (see NR 2nd edition 12.0.14, p498)
#
# numpy normalizes with 1/N on the inverse transform ifft,
# so we should normalize the freq-space representation with 1/sqrt(N).
- # But we're using the rfft, where N points are like N/2 complex points, so 1/sqrt(N/2)
+ # But we're using the rfft, where N points are like N/2 complex points,
+ # so 1/sqrt(N/2)
# So the power gets normalized by that twice and we have 2/N
#
# On top of this, the FFT assumes a sampling freq of 1 per second,
# and we want to preserve area under our curves.
# If our total time T = len(data)/freq is smaller than 1,
- # our df_real = freq/len(data) is bigger that the FFT expects (dt_fft = 1/len(data)),
+ # our df_real = freq/len(data) is bigger that the FFT expects
+ # (dt_fft = 1/len(data)),
# and we need to scale the powers down to conserve area.
# df_fft * F_fft(f) = df_real *F_real(f)
# F_real = F_fft(f) * (1/len)/(freq/len) = F_fft(f)*freq
# total_time = len(data)/float(freq)
# power *= 2.0 / float(freq)**2 / total_time
# power *= 2.0 / freq**2 * freq / len(data)
- power *= 2.0 / (freq * float(len(data)))
+ power *= 2.0 / (freq * _numpy.float(len(data)))
return (freq_axis, power)
+
def window_hann(length):
r"""Returns a Hann window array with length entries
.. math:: w_i = \frac{1}{2} (1-\cos(2\pi i/L))
"""
- win = zeros((length,), dtype=float)
+ win = _numpy.zeros((length,), dtype=_numpy.float)
for i in range(length):
- win[i] = 0.5*(1.0-cos(2.0*pi*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`.
If the number of samples in `data` is not a multiple of
`chunk_size`, we ignore the extra points.
"""
- assert chunk_size == floor_pow_of_two(chunk_size), \
- "chunk_size %d should be a power of 2" % chunk_size
+ 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 = linspace(0, freq/2, chunk_size/2+1)
+
+ win = window(chunk_size) # generate a window of the appropriate size
+ 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 = zeros((chunk_size/2+1,), dtype=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 = rfft(data[starti:stopi]*win)
- p_chunk = fft_chunk * fft_chunk.conj()
- power += p_chunk.astype(float)
- power /= float(nchunks)
+ 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)
return (freq_axis, power)
+
def unitary_avg_power_spectrum(data, freq=1.0, chunk_size=2048,
overlap=True, window=window_hann):
"""Compute the unitary average power spectrum of `data`.
--------
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
- power *= 2.0 / (freq*float(chunk_size)) * 8/3 # see unitary_power_spectrum()
- # * 8/3 to remove power from windowing
+ 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)) * 8.0 / 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).
# So our calulated power has and extra <w(t)**2> in it.
- # For the Hann window, <w(t)**2> = <0.5(1 + 2cos + cos**2)> = 1/4 + 0 + 1/8 = 3/8
- # For low frequency components, where the frequency of x(t) is ~= the frequency of w(t),
- # The normalization is not perfect. ??
+ # For the Hann window,
+ # <w(t)**2> = <0.5(1 + 2cos + cos**2)> = 1/4 + 0 + 1/8 = 3/8
+ # For low frequency components,
+ # where the frequency of x(t) is ~= the frequency of w(t),
+ # the normalization is not perfect. ??
# The normalization approaches perfection as chunk_size -> infinity.
return (freq_axis, power)
-
-class TestRFFT (unittest.TestCase):
+class TestRFFT (_unittest.TestCase):
r"""Ensure Numpy's FFT algorithm acts as expected.
Notes
.. math:: X_k = \sum_{m=0}^{n-1} x_m \exp^{-2\pi imk/n}
- .. [#dft] See the *Background information* section of :mod:`numpy.fft`.
+ .. [#dft] See the *Background information* section of
+ :py:mod:`numpy.fft`.
"""
def run_rfft(self, xs, Xs):
- i = complex(0,1)
+ i = _numpy.complex(0, 1)
n = len(xs)
Xa = []
for k in range(n):
- Xa.append(sum([x*exp(-2*pi*i*m*k/n) for x,m in zip(xs,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):
- assert (Xs[k]-Xa[k])/abs(Xa[k]) < 1e-6, \
- "rfft mismatch on element %d: %g != %g, relative error %g" \
- % (k, Xs[k], Xa[k], (Xs[k]-Xa[k])/abs(Xa[k]))
+ 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.
+ # SUM |x_m|^2 = 1/n SUM |X_k|^2.
# m=0 k=0
- timeSum = sum([abs(x)**2 for x in xs])
- freqSum = sum([abs(X)**2 for X in Xa])
- assert abs(freqSum/float(n) - timeSum)/timeSum < 1e-6, \
- "Mismatch on Parseval's, %g != 1/%d * %g" % (timeSum, n, freqSum)
+ 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):
- xs = [1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1]
- self.run_rfft(xs, rfft(xs))
+ "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 `unitary_rfft`.
+class TestUnitaryRFFT (_unittest.TestCase):
+ """Verify :py:func:`unitary_rfft`.
"""
- def run_unitary_rfft_parsevals(self, xs, freq, freqs, Xs):
+ def run_parsevals(self, xs, freq, freqs, Xs):
"""Check the discretized integral form of Parseval's theorem
Notes
.. 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]
- assert (df - 1/(len(xs)*dt))/df < 1e-6, \
- "Mismatch in spacing, %g != 1/(%d*%g)" % (df, len(xs), dt)
+ 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):
+ for k in range(len(Xs) - 1, 1, -1):
Xa.append(Xa[k])
- assert len(xs) == len(Xa), "Length mismatch %d != %d" % (len(xs), len(Xa))
- lhs = sum([abs(x)**2 for x in xs]) * dt
- rhs = sum([abs(X)**2 for X in Xa]) * df
- assert abs(lhs - rhs)/lhs < 1e-4, "Mismatch on Parseval's, %g != %g" \
- % (lhs, rhs)
-
- def test_unitary_rfft_parsevals(self):
+ 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 = pi
- freqs,Xs = unitary_rfft(xs, 1.0/dt)
- self.run_unitary_rfft_parsevals(xs, 1.0/dt, freqs, Xs)
-
+ 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.
\rect(t) = \begin{cases}
1& \text{if $|t| < 0.5$}, \\
0& \text{if $|t| \ge 0.5$}.
- \end{cases}
+ \end{cases}
"""
- if abs(t) < 0.5:
+ if _numpy.abs(t) < 0.5:
return 1
else:
return 0
-
- def run_unitary_rfft_rect(self, a=1.0, time_shift=5.0, samp_freq=25.6,
- samples=256):
- r"""Test `unitary_rttf` on known function `rect(at)`.
+
+ 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
-----
.. math:: \rfft(\rect(at)) = 1/|a|\cdot\sinc(f/a)
"""
- samp_freq = float(samp_freq)
- a = float(a)
-
- x = zeros((samples,), dtype=float)
- dt = 1.0/samp_freq
+ 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)
- #_test_unitary_rfft_parsevals(x, samp_freq, freq_axis, X)
-
+ 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 = complex(0.0,1.0) # sqrt(-1)
+ j = _numpy.complex(0.0, 1.0) # sqrt(-1)
for i in range(len(freq_axis)):
f = freq_axis[i]
- inverse_phase_shift = exp(j*2.0*pi*time_shift*f)
+ inverse_phase_shift = _numpy.exp(
+ j * 2.0 * _numpy.pi * time_shift * f)
X[i] *= inverse_phase_shift
-
- expected = zeros((len(freq_axis),), dtype=float)
+
+ 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
- assert sinc(0.5) == 2.0/pi, "abnormal sinc()"
+ 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/abs(a) * sinc(f/a)
-
+ expected[i] = 1.0 / _numpy.abs(a) * _numpy.sinc(f / a)
+
if TEST_PLOTS:
- pylab.figure()
- pylab.subplot(211)
- pylab.plot(arange(0, dt*samples, dt), x)
- pylab.title('time series')
- pylab.subplot(212)
- pylab.plot(freq_axis, X.real, 'r.')
- pylab.plot(freq_axis, X.imag, 'g.')
- pylab.plot(freq_axis, expected, 'b-')
- pylab.title('freq series')
-
- def test_unitary_rfft_rect(self):
+ 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_unitary_rfft_rect(a=0.5)
- self.run_unitary_rfft_rect(a=2.0)
- self.run_unitary_rfft_rect(a=0.7, samp_freq=50, samples=512)
- self.run_unitary_rfft_rect(a=3.0, samp_freq=60, samples=1024)
-
+ 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.
.. math:: \gaussian(a,t) = \exp^{-at^2}
"""
- return exp(-a * t**2)
-
- def run_unitary_rfft_gaussian(self, a=1.0, time_shift=5.0, samp_freq=25.6,
- samples=256):
- r"""Test `unitary_rttf` on known function `gaussian(a,t)`.
+ 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
-----
.. math::
- \rfft(\gaussian(a,t)) = \sqrt{\pi/a} \cdot \gaussian(1/a,\pi f)
+ \rfft(\gaussian(a,t))
+ = \sqrt{\pi/a} \cdot \gaussian(1/a,\pi f)
"""
- samp_freq = float(samp_freq)
- a = float(a)
-
- x = zeros((samples,), dtype=float)
- dt = 1.0/samp_freq
+ 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)
- #_test_unitary_rfft_parsevals(x, samp_freq, freq_axis, X)
-
+ 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 = complex(0.0,1.0) # sqrt(-1)
+ j = _numpy.complex(0.0, 1.0) # sqrt(-1)
for i in range(len(freq_axis)):
f = freq_axis[i]
- inverse_phase_shift = exp(j*2.0*pi*time_shift*f)
+ inverse_phase_shift = _numpy.exp(
+ j * 2.0 * _numpy.pi * time_shift * f)
X[i] *= inverse_phase_shift
-
- expected = zeros((len(freq_axis),), dtype=float)
+
+ expected = _numpy.zeros((len(freq_axis),), dtype=_numpy.float)
for i in range(len(freq_axis)):
f = freq_axis[i]
- expected[i] = sqrt(pi/a) * self.gaussian(1.0/a, pi*f) # see Wikipedia, or do the integral yourself.
-
+ # 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:
- pylab.figure()
- pylab.subplot(211)
- pylab.plot(arange(0, dt*samples, dt), x)
- pylab.title('time series')
- pylab.subplot(212)
- pylab.plot(freq_axis, X.real, 'r.')
- pylab.plot(freq_axis, X.imag, 'g.')
- pylab.plot(freq_axis, expected, 'b-')
- pylab.title('freq series')
-
- def test_unitary_rfft_gaussian(self):
+ 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_unitary_rfft_gaussian(a=0.5)
- self.run_unitary_rfft_gaussian(a=2.0)
- self.run_unitary_rfft_gaussian(a=0.7, samp_freq=50, samples=512)
- self.run_unitary_rfft_gaussian(a=3.0, samp_freq=60, samples=1024)
-
-class TestUnitaryPowerSpectrum (unittest.TestCase):
- def run_unitary_power_spectrum_sin(self, sin_freq=10, samp_freq=512,
- samples=1024):
- x = zeros((samples,), dtype=float)
- samp_freq = float(samp_freq)
+ 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] = sin(2.0 * pi * (i/samp_freq) * sin_freq)
- freq_axis, power = unitary_power_spectrum(x, samp_freq)
- imax = argmax(power)
-
- expected = zeros((len(freq_axis),), dtype=float)
- df = samp_freq/float(samples) # df = 1/T, where T = total_time
- i = int(sin_freq/df)
- # average power per unit time is
+ 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(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,
+ # 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 %g Hz of %g (%g, %g)" % \
- (sin_freq, expected[i], freq_axis[imax], power[imax])
- Pexp = 0
- P = 0
+
+ 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
- print " The total power should be %g (%g)" % (Pexp, P)
-
+ 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:
- pylab.figure()
- pylab.subplot(211)
- pylab.plot(arange(0, samples/samp_freq, 1.0/samp_freq), x, 'b-')
- pylab.title('time series')
- pylab.subplot(212)
- pylab.plot(freq_axis, power, 'r.')
- pylab.plot(freq_axis, expected, 'b-')
- pylab.title('%g samples of sin at %g Hz' % (samples, sin_freq))
-
- def test_unitary_power_spectrum_sin(self):
+ 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_unitary_power_spectrum_sin(sin_freq=5, samp_freq=512, samples=1024)
- self.run_unitary_power_spectrum_sin(sin_freq=5, samp_freq=512, samples=2048)
- self.run_unitary_power_spectrum_sin(sin_freq=5, samp_freq=512, samples=4098)
- self.run_unitary_power_spectrum_sin(sin_freq=7, samp_freq=512, samples=1024)
- self.run_unitary_power_spectrum_sin(sin_freq=5, samp_freq=1024, samples=2048)
- # finally, with some irrational numbers, to check that I'm not getting lucky
- self.run_unitary_power_spectrum_sin(sin_freq=pi, samp_freq=100*exp(1), samples=1024)
+ 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_unitary_power_spectrum_sin(sin_freq=5, samp_freq=512, samples=256)
-
- def run_unitary_power_spectrum_delta(self, amp=1, samp_freq=1,
- samples=256):
+ self.run_sin(sin_freq=5, samp_freq=512, samples=256)
+
+ def run_delta(self, amp=1, samp_freq=1, samples=256):
"""TODO
"""
- x = zeros((samples,), dtype=float)
- samp_freq = float(samp_freq)
+ 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)
-
+ 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)
# 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 = ones((len(freq_axis),), dtype=float) * expected_amp
-
- print "The power should be flat at y = %g (%g)" % (expected_amp, power[0])
-
+ 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:
- pylab.figure()
- pylab.subplot(211)
- pylab.plot(arange(0, samples/samp_freq, 1.0/samp_freq), x, 'b-')
- pylab.title('time series')
- pylab.subplot(212)
- pylab.plot(freq_axis, power, 'r.')
- pylab.plot(freq_axis, expected, 'b-')
- pylab.title('%g samples of delta amp %g' % (samples, amp))
-
- def _test_unitary_power_spectrum_delta(self):
+ 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"
- _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)
- _test_unitary_power_spectrum_delta(amp=1, samp_freq=0.5, samples=2048)# expected = 2*computed
- _test_unitary_power_spectrum_delta(amp=1, samp_freq=2.0, samples=2048)# expected = 0.5*computed
- _test_unitary_power_spectrum_delta(amp=3, samp_freq=1.0, samples=1024)
- _test_unitary_power_spectrum_delta(amp=pi, samp_freq=exp(1), samples=1024)
-
+ 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*sqrt(2.0*pi)) * exp(-0.5*((t-mean)/std)**2)
-
- def run_unitary_power_spectrum_gaussian(self, area=2.5, mean=5, std=1,
- samp_freq=10.24 ,samples=512):
+ 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 = zeros((samples,), dtype=float)
- mean = float(mean)
+ x = _numpy.zeros((samples,), dtype=_numpy.float)
+ mean = _numpy.float(mean)
for i in range(samples):
- t = i/float(samp_freq)
+ 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 self.gaussian() form to _gaussian(),
+ 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 _gaussian() above)
- # mean' = 0 (changing the mean in the time-domain just changes the phase in the freq-domain)
+ # 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*sqrt(2.0*pi))
- std = 1.0/(2.0*pi*std)
- expected = zeros((len(freq_axis),), dtype=float)
- df = float(samp_freq)/samples # 1/total_time ( = freq_axis[1]-freq_axis[0] = freq_axis[1])
+ 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
+ f = i * df
gaus = self.gaussian(area, mean, std, f)
- expected[i] = 2.0 * gaus**2 * samp_freq/samples
- print "The power should be a half-gaussian, ",
- print "with a peak at 0 Hz with amplitude %g (%g)" % (expected[0], power[0])
-
+ 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:
- pylab.figure()
- pylab.subplot(211)
- pylab.plot(arange(0, samples/samp_freq, 1.0/samp_freq), x, 'b-')
- pylab.title('time series')
- pylab.subplot(212)
- pylab.plot(freq_axis, power, 'r.')
- pylab.plot(freq_axis, expected, 'b-')
- pylab.title('freq series')
-
- def test_unitary_power_spectrum_gaussian(self):
+ 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,pi]:
- for std in [1,sqrt(2)]:
- for samp_freq in [10.0, exp(1)]:
- for samples in [1024,2048]:
- self.run_unitary_power_spectrum_gaussian(
+ 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_unitary_avg_power_spectrum_sin(self, sin_freq=10, samp_freq=512,
- samples=1024, chunk_size=512,
- overlap=True, window=window_hann):
+
+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 = zeros((samples,), dtype=float)
- samp_freq = float(samp_freq)
+ x = _numpy.zeros((samples,), dtype=_numpy.float)
+ samp_freq = _numpy.float(samp_freq)
for i in range(samples):
- x[i] = sin(2.0 * pi * (i/samp_freq) * sin_freq)
- freq_axis, power = unitary_avg_power_spectrum(x, samp_freq, chunk_size,
- overlap, window)
- imax = argmax(power)
-
- expected = zeros((len(freq_axis),), dtype=float)
- df = samp_freq/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 be a peak at %g Hz of %g (%g, %g)" % \
- (sin_freq, expected[i], freq_axis[imax], power[imax])
- Pexp = 0
- P = 0
+ 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
- print " The total power should be %g (%g)" % (Pexp, P)
-
+ P += power[i] * df
+ self.assertAlmostEqual(
+ Pexp, P, places,
+ 'The total power should be {} ({})'.format(Pexp, P))
+
if TEST_PLOTS:
- pylab.figure()
- pylab.subplot(211)
- pylab.plot(arange(0, samples/samp_freq, 1.0/samp_freq), x, 'b-')
- pylab.title('time series')
- pylab.subplot(212)
- pylab.plot(freq_axis, power, 'r.')
- pylab.plot(freq_axis, expected, 'b-')
- pylab.title('%g samples of sin at %g Hz' % (samples, sin_freq))
-
- def test_unitary_avg_power_spectrum_sin(self):
+ 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_unitary_avg_power_spectrum_sin(sin_freq=5, samp_freq=512, samples=1024)
- self.run_unitary_avg_power_spectrum_sin(sin_freq=5, samp_freq=512, samples=2048)
- self.run_unitary_avg_power_spectrum_sin(sin_freq=5, samp_freq=512, samples=4098)
- self.run_unitary_avg_power_spectrum_sin(sin_freq=17, samp_freq=512, samples=1024)
- self.run_unitary_avg_power_spectrum_sin(sin_freq=5, samp_freq=1024, samples=2048)
+ 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_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
- self.run_unitary_avg_power_spectrum_sin(sin_freq=pi, samp_freq=100*exp(1), samples=1024)
+ 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