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.
Starting with a DFT at time "n",
formula_1
The DFT for time "n" + 1 can be computed as
formula_2
and recursively thereafter as
formula_3
with
formula_4