pyfar.dsp.fft

The following documents the FFT functionality. More details and background is given in the FFT concepts.

Functions:

irfft(spec, n_samples, sampling_rate, fft_norm)	Calculate the IFFT of a single-sided Fourier spectrum.
normalization(spec, n_samples, sampling_rate)	Normalize a Fourier spectrum.
rfft(data, n_samples, sampling_rate, fft_norm)	Calculate the FFT of a real-valued time-signal.
rfftfreq(n_samples, sampling_rate)	Returns the positive discrete frequencies for which the FFT is calculated.

pyfar.dsp.fft.irfft(spec, n_samples, sampling_rate, fft_norm)

Calculate the IFFT of a single-sided Fourier spectrum.

The function takes only the right-hand side of the spectrum and returns a real-valued time signal. The normalization is considered according to 'fft_norm' as described in normalization and FFT concepts.

Parameters:

• spec (array, complex) – The complex valued right-hand side of the spectrum with dimensions (…, n_bins)

• n_samples (int) – The number of samples of the corresponding time signal. This is crucial to allow for the correct transform of time signals with an odd number of samples.

• sampling_rate (number) – sampling rate in Hz

• fft_norm ('none', 'unitary', 'amplitude', 'rms', 'power', 'psd') – See normalization.

Returns:

data – Array containing the time domain signal with dimensions (…, 'n_samples')

Return type:

array, double

pyfar.dsp.fft.normalization(spec, n_samples, sampling_rate, fft_norm='none', inverse=False, single_sided=True, window=None)

Normalize a Fourier spectrum.

Apply normalizations defined in [1] to the DFT spectrum. Note that the phase is maintained in all cases, i.e., instead of taking the squared absolute values for 'power' and 'psd', the complex spectra are multiplied with their absolute values to ensure a correct renormalization. For detailed information and explanations, refer to FFT concepts.

Parameters:

• spec (numpy array) – N dimensional array which has the frequency bins in the last dimension. E.g., spec.shape == (10,2,129) holds 10 times 2 spectra with 129 frequency bins each.

• n_samples (int) – number of samples of the corresponding time signal

• sampling_rate (number) – sampling rate of the corresponding time signal in Hz

• fft_norm (string, optional) –

  'none'

  Do not apply any normalization. Appropriate for energy signals such as impulse responses.

  'unitary'

  Multiply spec by factor of two as in [1] Eq. (8) (except for 0 Hz and the Nyquist frequency at half the sampling rate) to obtain the single-sided spectrum.

  'amplitude'

  Scale spectrum by 1/n_samples as in [1] Eq. (4) to obtain the amplitude spectrum.

  ’rms’

  Scale spectrum by \(1/\sqrt{2}\) as in [1] Eq.(10) to obtain the RMS spectrum.

  ’power’

  Power spectrum, which equals the squared RMS spectrum (except for the retained phase).

  ’psd’

  The power spectrum is scaled by n_samples/sampling_rate as in [1] Eq. (6)

  Note that the unitary normalization is also applied for amplitude, rms, power, and psd if the input spectrum is single sided (see single_sided).

• inverse (bool, optional) – apply the inverse normalization. The default is False.

• single_sided (bool, optional) – denotes if spec is a single sided spectrum up to half the sampling rate or a both sided (full) spectrum. If single_sided=True the unitary normalization according to [1] Eq. (8) is applied unless fft_norm='none'. The default is True.

• window (None, array like) – window that was applied to the time signal before performing the FFT. Affects the normalization as in [1] Eqs. (11-13). The window must be an array-like with n_samples length and. The default is None, which denotes that no window was applied.

Returns:

spec – normalized input spectrum

Return type:

numpy array

References

[1] (1,2,3,4,5,6,7)

J. Ahrens, C. Andersson, P. Höstmad, and W. Kropp, “Tutorial on Scaling of the Discrete Fourier Transform and the Implied Physical Units of the Spectra of Time-Discrete Signals,” Vienna, Austria, May 2020, p. e-Brief 600.

pyfar.dsp.fft.rfft(data, n_samples, sampling_rate, fft_norm)

Calculate the FFT of a real-valued time-signal.

The function returns only the right-hand side of the axis-symmetric spectrum. The normalization is considered according to 'fft_norm' as described in normalization and FFT concepts.

Parameters:

• data (array, double) – Array containing the time domain signal with dimensions (…, 'n_samples')

• n_samples (int) – The number of samples

• sampling_rate (number) – sampling rate in Hz

• fft_norm ('none', 'unitary', 'amplitude', 'rms', 'power', 'psd') – See documentation of normalization.

Returns:

spec – The complex valued right-hand side of the spectrum with dimensions (…, n_bins)

Return type:

array, complex

pyfar.dsp.fft.rfftfreq(n_samples, sampling_rate)

Returns the positive discrete frequencies for which the FFT is calculated.

If the number of samples \(N\) is even the number of frequency bins will be \(2/N+1\), if \(N\) is odd, the number of bins will be \((N+1)/2\).

Parameters:

• n_samples (int) – The number of samples in the signal

• sampling_rate (int) – The sampling rate of the signal

Returns:

frequencies – The positive discrete frequencies in Hz for which the FFT is calculated.

Return type:

array, double