Based on numpy.fft.rfft.
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)
+
+* :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
-# print time- and freq- space plots of the test transforms if True
TEST_PLOTS = False
-#TEST_PLOTS = True
-def floor_pow_of_two(num) :
- "Round num down to the closest exact a power of two."
+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 :
+ if int(lnum) != lnum:
num = 2**floor(lnum)
- return num
+ return int(num)
+
+def round_pow_of_two(num):
+ """Round `num` to the closest exact a power of two on a log scale.
-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 :
+ if int(lnum) != lnum:
num = 2**round(lnum)
- return num
+ return int(num)
+
+def ceil_pow_of_two(num):
+ """Round `num` up to the closest exact a power of two.
-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 :
+ if int(lnum) != lnum:
num = 2**ceil(lnum)
- return num
-
-def _test_rfft(xs, Xs) :
- # Numpy's FFT algoritm returns
- # n-1
- # X[k] = SUM x[m] exp (-j 2pi km /n)
- # m=0
- # (see http://www.tramy.us/numpybook.pdf)
- j = 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))]))
- 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)
+ return int(num)
-def _test_rfft_suite() :
- print "Test numpy rfft definition"
- xs = [1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1]
- _test_rfft(xs, rfft(xs))
+def unitary_rfft(data, freq=1.0):
+ """Compute the unitary 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]).
-
+ 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.
+ 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
freq_axis = linspace(0, freq/2, nsamps/2+1)
return (freq_axis, trans)
-def _test_unitary_rfft_parsevals(xs, freq, freqs, Xs):
- # Which should satisfy the discretized integral form of Parseval's theorem
- # n-1 n-1
- # SUM |x_m|^2 dt = SUM |X_k|^2 df
- # m=0 k=0
- dt = 1.0/freq
- df = freqs[1]-freqs[0]
- 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_suite():
- print "Test unitary rfft on Parseval's theorem"
- xs = [1,2,3,1,2,3,1,2,3,1,2,3,1,2,3,1]
- dt = 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 :
- return 1
- else :
- return 0
-
-def _test_unitary_rfft_rect(a=1.0, time_shift=5.0, samp_freq=25.6, samples=256) :
- "Show fft(rect(at)) = 1/abs(a) * 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] = _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_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 _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)
-
- x = zeros((samples,), dtype=float)
- dt = 1.0/samp_freq
- for i in range(samples) :
- t = i*dt
- x[i] = _gaussian(a, (t-time_shift))
- freq_axis, X = unitary_rfft(x, samp_freq)
- #_test_unitary_rfft_parsevals(x, samp_freq, freq_axis, X)
-
- # remove the phase due to our time shift
- j = 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) * _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"
- _test_unitary_rfft_gaussian(a=0.5)
- _test_unitary_rfft_gaussian(a=2.0)
- _test_unitary_rfft_gaussian(a=0.7, samp_freq=50, samples=512)
- _test_unitary_rfft_gaussian(a=3.0, samp_freq=60, samples=1024)
-
-
-
-def power_spectrum(data, freq=1.0) :
- """
- Compute the power spectrum of DATA taken with a sampling frequency FREQ.
- DATA must be real (not complex).
- Returns a tuple of two arrays, (freq_axis, power), suitable for plotting.
- If the number of samples in data is not an integer power of two,
+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))
power = trans * trans.conj() # 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):
+ """Compute the unitary power spectrum of the time series `data`.
+
+ See Also
+ --------
+ power_spectrum,unitary_avg_power_spectrum
+ """
freq_axis,power = power_spectrum(data, freq)
# One sided power spectral density, so 2|H(f)|**2 (see NR 2nd edition 12.0.14, p498)
#
return (freq_axis, power)
-def _test_unitary_power_spectrum_sin(sin_freq=10, samp_freq=512, samples=1024) :
- x = 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_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)
- # 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)
- 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_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
- _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 _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)
- x[i] = _gaussian2(area, mean, std, t)
- freq_axis, power = unitary_power_spectrum(x, samp_freq)
-
- # generate the predicted curve
- # by comparing our _gaussian2() form to _gaussian(),
- # we see that the Fourier transform of x(t) has parameters:
- # std' = 1/(2 pi std) (references declaring std' = 1/std are converting to angular frequency, not frequency like we are)
- # area' = area/[std sqrt(2*pi)] (plugging into FT of _gaussian() above)
- # mean' = 0 (changing the mean in the time-domain just changes the phase in the freq-domain)
- # So our power spectral density per unit time is given by
- # P(f) = 2 |X(f)|**2 / T
- # Where
- # T = samples/samp_freq (total time of data aquisition)
- mean = 0.0
- area = area /(std*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 = _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) :
- "Returns a Hann window array with length entries"
+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) :
+ 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 avg 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.
+ 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 :
+ if overlap:
chunk_step = chunk_size/2
- else :
+ else:
chunk_step = chunk_size
win = window(chunk_size) # generate a window of the appropriate size
# >>> 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) :
+ for i in range(nchunks):
starti = i*chunk_step
stopi = starti+chunk_size
fft_chunk = rfft(data[starti:stopi]*win)
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 unitary avgerage power spectrum of `data`.
+
+ See Also
+ --------
+ avg_power_spectrum,unitary_power_spectrum
"""
freq_axis,power = avg_power_spectrum(data, freq, chunk_size,
overlap, window)
# 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
- 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_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)
-
-
-def test() :
- _test_rfft_suite()
- _test_unitary_rfft_parsevals_suite()
- _test_unitary_rfft_rect_suite()
- _test_unitary_rfft_gaussian_suite()
- _test_unitary_power_spectrum_sin_suite()
- _test_unitary_power_spectrum_delta_suite()
- _test_unitary_power_spectrum_gaussian_suite()
- _test_unitary_avg_power_spectrum_sin_suite()
-
-if __name__ == "__main__" :
- if TEST_PLOTS :
- import pylab
- test()
- if TEST_PLOTS :
- pylab.show()
+
+
+class TestRFFT (unittest.TestCase):
+ r"""Ensure Numpy's FFT algorithm acts as expected.
+
+ Notes
+ -----
+
+ The expected return values are [#numpybook]_:
+
+ .. math:: X_k = \sum_{m=0}^{n-1} x_m \exp^{-j 2\pi k_m/n}
+
+ .. [#numpybook] http://www.tramy.us/numpybook.pdf
+ """
+ def run_rfft(self, xs, Xs):
+ j = 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))]))
+ 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 / FFT-tools.git / commitdiff