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

Starting with a DFT at time n,

X_{n}(k)=\sum _{m=0}^{N-1}x(n+m)e^{-j2\pi km/N}

The DFT for time n + 1 can be computed as

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

and recursively thereafter as

X_{n+1}(k)=e^{j2\pi k/N}\left[X_{n}(k)+\Delta \right]

with

\Delta =x(n+N)-x(n)

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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.