NuRadioReco.utilities.fft module

A wrapper around the numpy fft routines to achive a coherent normalization of the fft

As we have real valued data in the time domain, we use the ‘real ffts’ that omit the negative frequencies in Fourier space. To account for the missing power in the frequency domain, we multiply the frequency spectrum by sqrt(2), and divide the iFFT with 1/sqrt(2) accordingly. The frequency spectrum is divided by the sampling rate so that the units if the spectrum are volts/GHz instead of volts/bin and the voltage in the frequency domain is independent of the sampling rate.

Then, a calculation of the energy leads the same result in the time and frequency domain, i.e.

np.sum(trace**2) * dt = np.sum(spectrum**2) * df

with units of \(V^2/GHz\). In order to obtain the correct units (i.e. energy in eV) one additionally has to divide by the impedance Z.

Note that in our convention, the above equality holds only approximately, as also the zero-frequency and the Nyquist frequency are multiplied by sqrt(2). This effect is however very small and negligible in practice.

Finally, we remark that our normalization ensures implies the FFT describes the energy spectral density; this is in contrast to another common convention where the Fourier transform describes the power spectral density. One should keep this in mind especially when working with e.g. noise temperatures which are defined using the latter convention.

NuRadioReco.utilities.fft.freqs(num_samples, sampling_rate)

Returns frequency bins for FFT.

Parameters:

num_samples: int

The number of sample in the time domain

sampling_rate: float

Sampling rate of the trace

Returns:

frequencies: np.array

Frequency binning

NuRadioReco.utilities.fft.time2freq(trace, sampling_rate)

Performs forward FFT with correct normalization that conserves the power.

Parameters:

trace: np.array

time trace to be transformed into frequency space

sampling_rate: float

sampling rate of the trace

Returns:

spec: np.array

Frequency spectrum

NuRadioReco.utilities.fft.freq2time(spectrum, sampling_rate, n=None)

Performs backward FFT with correct normalization that conserves the power.

Parameters:

spectrum: complex np array

the frequency spectrum

sampling_rate: float

sampling rate of the spectrum

n: int

the number of sample in the time domain (relevant if time trace has an odd number of samples)

Returns:

trace: np.array

Time series/trace