Sliding DFT
{{Multiple issues|
{{refimprove|date=February 2020}}
{{Context|date=February 2020}}
}}
In applied mathematics, the sliding discrete Fourier transform is a recursive algorithm to compute successive STFTs of input data frames that are a single sample apart (hopsize − 1).{{cite journal |last1=Bradford |first1=Russell |title=SLIDING IS SMOOTHER THAN JUMPING |journal=Proceedings ICMC 2005 |date=2005 |url=http://www.music.mcgill.ca/~ich/research/misc/papers/cr1137.pdf}} The calculation for the sliding DFT is closely related to Goertzel algorithm.{{cn|date=January 2023}}
Definition
Assuming that the hopsize between two consecutive DFTs is 1 sample, then{{cite book |last1=Lazzarini |first1=Victor |title=Spectral Music Design | publisher=Oxford Univ. Press |date=2021 }}
\begin{align}
F_{t+1}(n) &= \sum_{k=0}^{N-1} f_{k+t+1}e^{-j2\pi k n/N}\\
&= \sum_{m=1}^N f_{m+t}e^{-j2\pi (m-1) n/N} \\
&= e^{j2\pi n/N} \left[ \sum_{m=0}^{N-1} f_{m+t}e^{-j2\pi m n /N} - f_t + f_{t+N} \right] \\
&= e^{j2\pi n/N} \left[F_t(n) - f_t + f_{t+N} \right].
\end{align}
From this definition above, the DFT can be computed recursively thereafter. However, implementing the window function on a sliding DFT is difficult due to its recursive nature, therefore it is done exclusively in a frequency domain.{{cite journal |last1=Rafii |first1=Zafar |title=Sliding Discrete Fourier Transform with Kernel Windowing |journal=IEEE Signal Processing Magazine |date=14 November 2018 |volume=35 |issue=6 |doi=10.1109/MSP.2018.2855727 |url=https://ieeexplore.ieee.org/document/8498079}}
= Sliding windowed infinite Fourier transform =
It is not possible to implement asymmetric window functions into sliding DFT. However, the IIR version called sliding windowed infinite Fourier transform (SWIFT) provides an exponential window and the αSWIFT calculates two sDFTs in parallel where slow-decaying one is subtracted by fast-decaying one, therefore a window function of .{{cite magazine
| last1 = Grado | first1 = Logan L.
| last2 = Johnson | first2 = Matthew D.
| last3 = Netoff | first3 = Theoden I.
| date = September 2017
| doi = 10.1109/msp.2017.2718039
| issue = 5
| magazine = IEEE Signal Processing Magazine
| pages = 183–188
| publisher = Institute of Electrical and Electronics Engineers
| title = Tips & Tricks: The Sliding Windowed Infinite Fourier Transform
| url = https://par.nsf.gov/servlets/purl/10078055
| volume = 34}}