From aff37ad2d5786e87ba4f5dc605549bffde8cdc2e Mon Sep 17 00:00:00 2001 From: "W. Trevor King" Date: Mon, 17 May 2010 18:38:33 -0400 Subject: [PATCH] Converted hooke.util.fft to Hooke coding style. * Add numpydoc docstrings (to most functions) * Converted hackish test suites to hackish `unittest.TestCase`s. Still ugly, but now `nosetests` finds them. They should really use .assertAlmostEqual and company... * Added neccessary function definitions to doc/conf.py for the 'math' sections of fft's docstrings. --- FFT_tools.py | 1012 +++++++++++++++++++++++++++++--------------------- 1 file changed, 582 insertions(+), 430 deletions(-) diff --git a/FFT_tools.py b/FFT_tools.py index b79d3b4..107c216 100644 --- a/FFT_tools.py +++ b/FFT_tools.py @@ -40,82 +40,102 @@ Provides a unitary discrete FFT and a windowed version. 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 @@ -153,138 +173,29 @@ def unitary_rfft(data, freq=1.0) : 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)) @@ -296,7 +207,13 @@ def power_spectrum(data, freq=1.0) : 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) # @@ -326,195 +243,68 @@ def unitary_power_spectrum(data, freq=1.0) : 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 = - # 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 = = (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 @@ -523,7 +313,7 @@ def avg_power_spectrum(data, freq=1.0, chunk_size=2048, # >>> 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) @@ -533,9 +323,12 @@ def avg_power_spectrum(data, freq=1.0, chunk_size=2048, 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) @@ -551,67 +344,426 @@ def unitary_avg_power_spectrum(data, freq=1.0, chunk_size=2048, # 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 = + # 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 = = (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) -- 2.26.2