Sliding DFT

This article has multiple issues. Please help improve it or discuss these issues on the talk page. (Learn how and when to remove these messages) This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Sliding DFT" – news · newspapers · books · scholar · JSTOR (February 2020) (Learn how and when to remove this message) This article provides insufficient context for those unfamiliar with the subject. Please help improve the article by providing more context for the reader. (February 2020) (Learn how and when to remove this message) (Learn how and when to remove this message)

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).^[1] The calculation for the sliding DFT is closely related to Goertzel algorithm.^{[citation needed]}

Assuming that the hopsize between two consecutive DFTs is 1 sample, then^[2]

$${\displaystyle {\begin{aligned}F_{t+1}(n)&=\sum _{k=0}^{N-1}f_{k+t+1}e^{-j2\pi kn/N}\\&=\sum _{m=1}^{N}f_{m+t}e^{-j2\pi (m-1)n/N}\\&=e^{j2\pi n/N}\left[\sum _{m=0}^{N-1}f_{m+t}e^{-j2\pi mn/N}-f_{t}+f_{t+N}\right]\\&=e^{j2\pi n/N}\left[F_{t}(n)-f_{t}+f_{t+N}\right].\end{aligned}}}$$

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.^[3]

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 ${\displaystyle w(x)=e^{-x\alpha }-e^{-x\beta }}$.^[4]

This signal processing-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e