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).[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

Bradford, Russell (2005). "SLIDING IS SMOOTHER THAN JUMPING" (PDF). Proceedings ICMC 2005.

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[1] Bradford, Russell (2005). "SLIDING IS SMOOTHER THAN JUMPING" (PDF). Proceedings ICMC 2005.