+++ /dev/null
-# Copyright (C) 2010-2012 W. Trevor King <wking@tremily.us>
-#
-# This file is part of Hooke.
-#
-# Hooke is free software: you can redistribute it and/or modify it under the
-# terms of the GNU Lesser General Public License as published by the Free
-# Software Foundation, either version 3 of the License, or (at your option) any
-# later version.
-#
-# Hooke 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 Lesser General Public License for more
-# details.
-#
-# You should have received a copy of the GNU Lesser General Public License
-# along with Hooke. If not, see <http://www.gnu.org/licenses/>.
-
-"""Wrap :mod:`numpy.fft` to produce 1D unitary transforms and power spectra.
-
-Define some FFT wrappers to reduce clutter.
-Provides a unitary discrete FFT and a windowed version.
-Based on :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`
-"""
-
-import unittest
-
-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
-
-
-TEST_PLOTS = False
-
-def floor_pow_of_two(num):
- """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 = log2(num)
- if int(lnum) != lnum:
- num = 2**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.
-
- 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 = log2(num)
- if int(lnum) != lnum:
- num = 2**round(lnum)
- return int(num)
-
-def ceil_pow_of_two(num):
- """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 = log2(num)
- if int(lnum) != lnum:
- num = 2**ceil(lnum)
- return int(num)
-
-def unitary_rfft(data, freq=1.0):
- """Compute the unitary Fourier transform of real data.
-
- Unitary = preserves power [Parseval's theorem].
-
- Parameters
- ----------
- data : iterable
- Real (not complex) data taken with a sampling frequency `freq`.
- freq : real
- Sampling frequency.
-
- 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.
- """
- 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.
- # 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
- # which has the discrete form
- # n-1 n-1
- # SUM |x_m|^2 dt = SUM |X'_k|^2 df
- # m=0 k=0
- # with X'_k = AX, this gives us
- # n-1 n-1
- # SUM |x_m|^2 = A^2 df/dt SUM |X'_k|^2
- # m=0 k=0
- # so we see
- # A^2 df/dt = 1/n
- # A^2 = 1/n dt/df
- # From Numerical Recipes (http://www.fizyka.umk.pl/nrbook/bookcpdf.html),
- # Section 12.1, we see that for a sampling rate dt, the maximum frequency
- # f_c in the transformed data is the Nyquist frequency (12.1.2)
- # f_c = 1/2dt
- # and the points are spaced out by (12.1.5)
- # df = 1/ndt
- # so
- # dt = 1/ndf
- # dt/df = 1/ndf^2
- # A^2 = 1/n^2df^2
- # A = 1/ndf = ndt/n = dt
- # 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)
- return (freq_axis, trans)
-
-def power_spectrum(data, freq=1.0):
- """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 = 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.
- return (freq_axis, power)
-
-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)
- #
- # 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)
- # 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)),
- # 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
- # So the power gets normalized by *that* twice and we have 2/N * freq**2
-
- # power per unit time
- # measure x(t) for time T
- # X(f) = int_0^T x(t) exp(-2 pi ift) dt
- # PSD(f) = 2 |X(f)|**2 / T
-
- # 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)))
-
- return (freq_axis, power)
-
-def window_hann(length):
- r"""Returns a Hann window array with length entries
-
- Notes
- -----
- The Hann window with length :math:`L` is defined as
-
- .. math:: w_i = \frac{1}{2} (1-\cos(2\pi i/L))
- """
- win = zeros((length,), dtype=float)
- for i in range(length):
- win[i] = 0.5*(1.0-cos(2.0*pi*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`.
-
- 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.
- """
- assert chunk_size == floor_pow_of_two(chunk_size), \
- "chunk_size %d should be a power of 2" % chunk_size
-
- nchunks = len(data)/chunk_size # integer division = implicit floor
- if overlap:
- 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)
- # 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)
- 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)
- 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`.
-
- 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
- power *= 2.0 / (freq*float(chunk_size)) * 8/3 # see unitary_power_spectrum()
- # * 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. ??
- # The normalization approaches perfection as chunk_size -> infinity.
- return (freq_axis, power)
-
-
-
-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 :mod:`numpy.fft`.
- """
- def run_rfft(self, xs, Xs):
- i = 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))]))
- 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]))
- # 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([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)
-
- 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))
-
-class TestUnitaryRFFT (unittest.TestCase):
- """Verify `unitary_rfft`.
- """
- def run_unitary_rfft_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]
- assert (df - 1/(len(xs)*dt))/df < 1e-6, \
- "Mismatch in spacing, %g != 1/(%d*%g)" % (df, len(xs), dt)
- Xa = list(Xs)
- 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):
- "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)
-
- 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 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)`.
-
- Notes
- -----
- Analytic result:
-
- .. 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
- 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)
-
- # remove the phase due to our time shift
- j = 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)
- X[i] *= inverse_phase_shift
-
- expected = zeros((len(freq_axis),), dtype=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()"
- for i in range(len(freq_axis)):
- f = freq_axis[i]
- expected[i] = 1.0/abs(a) * 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):
- "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)
-
- def gaussian(self, a, t):
- r"""Gaussian function.
-
- Notes
- -----
-
- .. 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)`.
-
- Notes
- -----
- Analytic result:
-
- .. math::
-
- \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
- 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)
-
- # remove the phase due to our time shift
- j = 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)
- X[i] *= inverse_phase_shift
-
- expected = zeros((len(freq_axis),), dtype=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.
-
- 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):
- "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)
- 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
- # 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 %g Hz of %g (%g, %g)" % \
- (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 %g (%g)" % (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):
- "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)
- # 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):
- """TODO
- """
- x = zeros((samples,), dtype=float)
- samp_freq = 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 = ones((len(freq_axis),), dtype=float) * expected_amp
-
- print "The power should be flat at y = %g (%g)" % (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):
- "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)
-
- 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):
- """TODO.
- """
- x = zeros((samples,), dtype=float)
- mean = float(mean)
- for i in range(samples):
- t = i/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(),
- # 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*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])
- 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
- print "The power should be a half-gaussian, ",
- print "with a peak at 0 Hz with amplitude %g (%g)" % (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):
- "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(
- 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):
- """TODO
- """
- x = zeros((samples,), dtype=float)
- samp_freq = 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
- 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)
-
- 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):
- "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)
- # 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)
projects / hooke.git / commitdiff