Sliding DFT

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).

Definition

Assuming that the hopsize between two consecutive DFTs is 1 sample, then

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

From this definition, the DFT can be computed recursively thereafter.

References

Jacobsen, E., Lyons, R.: ‘The sliding DFT’, IEEE Signal Process. Mag., 2013, 20, (2), pp. 74–80

Jacobsen, E., Lyons, R.: ‘An update to the sliding DFT’, IEEE Signal Process. Mag., 2014, 21, (1), pp. 110–111