irfft

The inverse fast Fourier transform (irfft) is a computational algorithm used to convert a sequence of values from the frequency domain back into the time domain. It is the inverse operation of the fast Fourier transform (FFT), which transforms time-domain signals into their frequency-domain representations. The irfft function is particularly useful in applications requiring efficient signal processing, such as audio synthesis, image reconstruction, and solving partial differential equations.

The irfft function differs from the standard inverse FFT in that it assumes the input is a one-sided spectrum, meaning it only contains the positive frequencies of the signal. This is common in real-valued signals, where negative frequencies are redundant due to the Hermitian symmetry property. By leveraging this property, irfft reduces computational overhead by avoiding redundant calculations of negative frequencies.

Mathematically, the irfft function computes the inverse transform of a given frequency-domain sequence, typically represented as a complex-valued array. The output is a real-valued time-domain signal, which may include a linear phase component if the input spectrum was not centered at zero frequency. The function often includes an optional parameter to adjust for this phase shift, ensuring the reconstructed signal aligns correctly with its original time-domain form.

Implementations of irfft are available in many scientific computing libraries, such as NumPy in Python, where it is provided as `numpy.irfft`. This function efficiently handles large datasets and supports additional options like output length and axis specification for multi-dimensional signals. The algorithm’s efficiency stems from its reliance on the FFT’s divide-and-conquer approach, which minimizes the number of operations required for the inverse transformation.