invFFx

invFFx is the inverse operation of a parametric class of frequency-domain transforms referred to as FFx. FFx denotes a family of linear transforms that map data from the time or spatial domain into a frequency-like domain, with the parameter x controlling properties such as windowing, phase, or scaling. The term invFFx therefore represents the process of recovering the original signal from its FFx-transformed representation, provided FFx is invertible.

Definition and properties: If FFx is represented by a square invertible matrix F_x, the forward transform is y = F_x x, and the inverse transform is x = F_x^{-1} y. In the discrete setting, invFFx is implemented by applying the inverse matrix or by an algorithm aligned with the chosen FFx definition. When FFx specializes to the discrete Fourier transform, invFFx coincides with the inverse DFT. If FFx reduces to the standard FFT, invFFx corresponds to the inverse FFT.

Computational aspects and applications: The cost of invFFx depends on the specific FFx family; many implementations reuse fast algorithms with O(n log n) complexity for large n. invFFx is used in signal reconstruction, deconvolution, and image processing tasks where a frequency-domain representation is manipulated and then mapped back to the original domain. Important considerations include the invertibility of FFx, numerical stability, and handling of imperfections such as noise or incomplete data, which may require regularization or the use of a pseudoinverse.