FFT_tools

Wrap Numpy’s fft module to reduce clutter.

Provides a unitary discrete FFT and a windowed version based on numpy.fft.rfft().

Main entry functions:

Functions

avg_power_spectrum(data[, freq, chunk_size, ...]) Compute the avgerage power spectrum of data.
ceil_pow_of_two(num) Round num up to the closest exact a power of two.
floor_pow_of_two(num) Round num down to the closest exact a power of two.
power_spectrum(data[, freq]) Compute the power spectrum of the time series data.
round_pow_of_two(num) Round num to the closest exact a power of two on a log scale.
unitary_avg_power_spectrum(data[, freq, ...]) Compute the unitary average power spectrum of data.
unitary_power_spectrum(data[, freq]) Compute the unitary power spectrum of the time series data.
unitary_rfft(data[, freq]) Compute the unitary Fourier transform of real data.
window_hann(length) Returns a Hann window array with length entries

Classes

TestRFFT([methodName]) Ensure Numpy’s FFT algorithm acts as expected.
TestUnitaryAvgPowerSpectrum([methodName])
TestUnitaryPowerSpectrum([methodName])
TestUnitaryRFFT([methodName]) Verify unitary_rfft().
class FFT_tools.TestRFFT(methodName='runTest')

Ensure Numpy’s FFT algorithm acts as expected.

The expected return values are [1]:

X_k = \sum_{m=0}^{n-1} x_m \exp^{-2\pi imk/n}

[1]See the Background information section of numpy.fft.
test_rfft()

Test NumPy’s builtin numpy.fft.rfft()

class FFT_tools.TestUnitaryRFFT(methodName='runTest')

Verify unitary_rfft().

gaussian(a, t)

Gaussian function.

\gaussian(a,t) = \exp^{-at^2}

rect(t)

Rectangle function.

\rect(t) = \begin{cases} 1& \text{if $|t| < 0.5$}, \\ 0& \text{if $|t| \ge 0.5$}. \end{cases}

run_gaussian(a=1.0, time_shift=5.0, samp_freq=25.6, samples=256)

Test unitary_rttf() on known function gaussian().

Analytic result:

\rfft(\gaussian(a,t)) = \sqrt{\pi/a} \cdot \gaussian(1/a,\pi f)

run_parsevals(xs, freq, freqs, Xs)

Check the discretized integral form of Parseval’s theorem

Which is:

\sum_{m=0}^{n-1} |x_m|^2 dt = \sum_{k=0}^{n-1} |X_k|^2 df

run_rect(a=1.0, time_shift=5.0, samp_freq=25.6, samples=256)

Test unitary_rfft() on known function rect().

Analytic result:

\rfft(\rect(at)) = 1/|a|\cdot\sinc(f/a)

test_gaussian()

Test unitary FFTs on variously shaped gaussian functions.

test_parsevals()

Test unitary rfft on Parseval’s theorem

test_rect()

Test unitary FFTs on variously shaped rectangular functions.

FFT_tools.avg_power_spectrum(data, freq=1.0, chunk_size=2048, overlap=True, window=<function window_hann>)

Compute the avgerage power spectrum of data.

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.
freq_axis,power : numpy.ndarray
Arrays ready for plotting.

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.

FFT_tools.ceil_pow_of_two(num)

Round num up to the closest exact a power of two.

>>> ceil_pow_of_two(3)
4
>>> ceil_pow_of_two(11)
16
>>> ceil_pow_of_two(15)
16
FFT_tools.floor_pow_of_two(num)

Round num down to the closest exact a power of two.

>>> floor_pow_of_two(3)
2
>>> floor_pow_of_two(11)
8
>>> floor_pow_of_two(15)
8
FFT_tools.power_spectrum(data, freq=1.0)

Compute the power spectrum of the time series data.

data : iterable
Real (not complex) data taken with a sampling frequency freq.
freq : real
Sampling frequency.
freq_axis,power : numpy.ndarray
Arrays ready for plotting.

If the number of samples in data is not an integer power of two, the FFT ignores some of the later points.

unitary_power_spectrum,avg_power_spectrum

FFT_tools.round_pow_of_two(num)

Round num to the closest exact a power of two on a log scale.

>>> round_pow_of_two(2.9) # Note rounding on *log scale*
4
>>> round_pow_of_two(11)
8
>>> round_pow_of_two(15)
16
FFT_tools.unitary_avg_power_spectrum(data, freq=1.0, chunk_size=2048, overlap=True, window=<function window_hann>)

Compute the unitary average power spectrum of data.

avg_power_spectrum,unitary_power_spectrum

FFT_tools.unitary_power_spectrum(data, freq=1.0)

Compute the unitary power spectrum of the time series data.

power_spectrum,unitary_avg_power_spectrum

FFT_tools.unitary_rfft(data, freq=1.0)

Compute the unitary Fourier transform of real data.

Unitary = preserves power [Parseval’s theorem].

data : iterable
Real (not complex) data taken with a sampling frequency freq.
freq : real
Sampling frequency.
freq_axis,trans : numpy.ndarray
Arrays ready for plotting.

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.

FFT_tools.window_hann(length)

Returns a Hann window array with length entries

The Hann window with length L is defined as

w_i = \frac{1}{2} (1-\cos(2\pi i/L))