Sliding DFT
STFT with sample-by-sample computation
In digital signal processing, the sliding discrete Fourier transform (sliding DFT or SDFT) is a recursive algorithm for computing a sequence of discrete Fourier transforms on overlapping data windows. Successive windows differ by only one sample, so the transform for the new window can be obtained by updating the transform of the previous window rather than recomputing it from scratch.
The sliding DFT is useful in applications that require a new spectral estimate for every input sample, especially when only a subset of DFT frequency bins are needed. In these cases, the sliding DFT can require fewer computations than repeatedly evaluating a complete DFT or FFT. The sliding DFT's recursive structure is closely related to resonator based DFT methods such as the Goertzel algorithm.
Definition
Assuming that the hop size between two consecutive DFTs is one sample, then
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{\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. For each frequency bin calculated, the recurrence updates the previous DFT value when a new input sample arrives, rather than needing to recompute the transform over the entire data frame. This makes the sliding DFT useful for applications requiring a new spectral estimate for every input sample, particularly when only a subset of the DFT bins is required. In such applications it can require fewer computations than repeatedly applying a FFT, for instance.
Implementation and computational cost
A single DFT bin of the sliding DFT can be implemented as a comb filter followed by a complex resonator.
Begin with the source’s own compact description: “Sliding DFT” is sTFT with sample-by-sample computation. The dossier treats that line as a proposition to test through Sliding, STFT and sample-by-sample, not as a finished interpretation.
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