-# Copyright (C) 2008-2011 W. Trevor King
+# 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
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
+
Main entry functions:
- unitary_rfft(data, freq=1.0)
- power_spectrum(data, freq=1.0)
- unitary_power_spectrum(data, freq=1.0)
- avg_power_spectrum(data, freq=1.0, chunk_size=2048, overlap=True, window=window_hann)
- unitary_avg_power_spectrum(data, freq=1.0, chunk_size=2048, overlap=True, window=window_hann)
+
+>>> 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)
"""
-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
+import numpy as _numpy
__version__ = '0.3'
TEST_PLOTS = False
-def floor_pow_of_two(num) :
+def floor_pow_of_two(num):
"Round num down to the closest exact a power of two."
- lnum = log2(num)
- if int(lnum) != lnum :
- num = 2**floor(lnum)
+ lnum = _numpy.log2(num)
+ if int(lnum) != lnum:
+ num = 2**_numpy.floor(lnum)
return num
-def round_pow_of_two(num) :
+def round_pow_of_two(num):
"Round num to the closest exact a power of two on a log scale."
- lnum = log2(num)
- if int(lnum) != lnum :
- num = 2**round(lnum)
+ lnum = _numpy.log2(num)
+ if int(lnum) != lnum:
+ num = 2**_numpy.round(lnum)
return num
-def ceil_pow_of_two(num) :
+def ceil_pow_of_two(num):
"Round num up to the closest exact a power of two."
- lnum = log2(num)
- if int(lnum) != lnum :
- num = 2**ceil(lnum)
+ lnum = _numpy.log2(num)
+ if int(lnum) != lnum:
+ num = 2**_numpy.ceil(lnum)
return num
-def _test_rfft(xs, Xs) :
+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 = complex(0,1)
+ j = _numpy.complex(0,1)
n = len(xs)
Xa = []
- for k in range(n) :
- Xa.append(sum([x*exp(-j*2*pi*k*m/n) for x,m in zip(xs,range(n))]))
+ 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):
- 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]))
+ 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.
+ # 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_suite() :
- print "Test numpy rfft definition"
+ 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))
+
+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, rfft(xs))
+ _test_rfft(xs, _numpy.fft.rfft(xs))
+
+def unitary_rfft(data, freq=1.0):
+ """Compute the Fourier transform of real data.
-def unitary_rfft(data, freq=1.0) :
- """
- Compute the Fourier transform of real data.
Unitary (preserves power [Parseval's theorem]).
-
+
If the units on your data are Volts,
and your sampling frequency is in Hz,
then freq_axis will be in Hz,
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 _test_unitary_rfft_parsevals(xs, freq, freqs, Xs):
# m=0 k=0
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)
+ 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) :
+ 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)
+ 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"
+ 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 = pi
+ 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 abs(t) < 0.5 :
+def _rect(t):
+ if _numpy.abs(t) < 0.5:
return 1
- else :
+ 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) * sinc(f/a)"
- samp_freq = float(samp_freq)
- a = float(a)
+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 = zeros((samples,), dtype=float)
+ x = _numpy.zeros((samples,), dtype=_numpy.float)
dt = 1.0/samp_freq
- for i in range(samples) :
+ 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 = complex(0.0,1.0) # sqrt(-1)
- for i in range(len(freq_axis)) :
+ 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()"
- for i in range(len(freq_axis)) :
+ 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/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_suite() :
- print "Test unitary FFTs on variously shaped rectangular functions"
+ 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 exp(-a * t**2)
+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) :
+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 = float(samp_freq)
- a = float(a)
+ samp_freq = _numpy.float(samp_freq)
+ a = _numpy.float(a)
- x = zeros((samples,), dtype=float)
+ x = _numpy.zeros((samples,), dtype=_numpy.float)
dt = 1.0/samp_freq
- for i in range(samples) :
+ 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 = complex(0.0,1.0) # sqrt(-1)
- for i in range(len(freq_axis)) :
+ 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)
- for i in range(len(freq_axis)) :
+ 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) * _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_suite() :
- print "Test unitary FFTs on variously shaped gaussian functions"
+ # 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)
-def power_spectrum(data, freq=1.0) :
- """
- Compute the power spectrum of DATA taken with a sampling frequency FREQ.
+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,
the FFT ignores some of the later points.
"""
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])
+ 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) :
+def unitary_power_spectrum(data, freq=1.0):
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 _test_unitary_power_spectrum_sin(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)
+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 = argmax(power)
+ imax = _numpy.argmax(power)
- expected = zeros((len(freq_axis),), dtype=float)
- df = samp_freq/float(samples) # df = 1/T, where T = total_time
+ 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
+ # 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(f0) = 0.5/df
# where f0 is the sin's frequency
#
- # or :
+ # 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)
# 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])
+ 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)) :
+ 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_suite() :
- print "Test unitary power spectrums on variously shaped sin functions"
+ 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)
- # finally, with some irrational numbers, to check that I'm not getting lucky
- _test_unitary_power_spectrum_sin(sin_freq=pi, samp_freq=100*exp(1), samples=1024)
+ # 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 = zeros((samples,), dtype=float)
- samp_freq = float(samp_freq)
+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)
# 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_suite() :
- print "Test unitary power spectrums on various delta functions"
+ 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)
- _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
+ # 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=pi, samp_freq=exp(1), samples=1024)
+ _test_unitary_power_spectrum_delta(
+ amp=_numpy.pi, samp_freq=_numpy.exp(1), samples=1024)
-def _gaussian2(area, mean, std, t) :
+def _gaussian2(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 _test_unitary_power_spectrum_gaussian(area=2.5, mean=5, std=1, samp_freq=10.24 ,samples=512) : #1024
- x = zeros((samples,), dtype=float)
- mean = float(mean)
- for i in range(samples) :
- t = i/float(samp_freq)
+ 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)
+ # 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)
+ # 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)) :
+ 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, ",
- 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_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=pi, std=sqrt(2), samp_freq=exp(1), samples=1024)
-
-def window_hann(length) :
+ 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):
"Returns a Hann window array with length entries"
- win = zeros((length,), dtype=float)
- for i in range(length) :
- win[i] = 0.5*(1.0-cos(2.0*pi*float(i)/(length)))
+ 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)))
# 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 avg power spectrum of DATA taken with a sampling frequency FREQ.
+ 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.
"""
- 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
- if overlap :
+ if overlap:
chunk_step = chunk_size/2
- else :
+ 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)
+ 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)
- for i in range(nchunks) :
+ 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)
+ fft_chunk = _numpy.fft.rfft(data[starti:stopi]*win)
p_chunk = (fft_chunk * fft_chunk.conj()).real
- power += p_chunk.astype(float)
- power /= float(nchunks)
+ 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 average power spectrum, preserving normalization
+ overlap=True, window=window_hann):
+ """Compute the average power spectrum, preserving normalization
"""
- 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)) * 8/3
+ # * 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)
-def _test_unitary_avg_power_spectrum_sin(sin_freq=10, samp_freq=512, samples=1024,
- chunk_size=512, overlap=True,
- window=window_hann) :
- 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
+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 be a peak at %g Hz of %g (%g, %g)" % \
- (sin_freq, expected[i], freq_axis[imax], power[imax])
+ 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)) :
+ 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_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)
+ 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=pi, samp_freq=100*exp(1), samples=1024)
+ _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() :
+def test():
_test_rfft_suite()
_test_unitary_rfft_parsevals_suite()
_test_unitary_rfft_rect_suite()
_test_unitary_power_spectrum_gaussian_suite()
_test_unitary_avg_power_spectrum_sin_suite()
-if __name__ == "__main__" :
+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',
+ 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 pylab
+ import matplotlib.pyplot as _pyplot
TEST_PLOTS = True
test()
if options.plot:
- pylab.show()
+ _pyplot.show()
projects / FFT-tools.git / blobdiff