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
F_(n) &= \sum_^ f_e^\\
&= \sum_^N f_e^ \\
&= e^ \left \sum_^ f_e^ - f_t + f_ \right\\
&= e^ \left _t(n) - f_t + f_ \right
\end

From this definition, the DFT can be computed recursively thereafter. However, implementing the window function on a sliding DFT is difficult due to its recursive nature, therefore it is done exclusively in a frequency domain.

There is an alternate way to implement the sliding DFT, called sliding Goertzel algorithm, which does the same except it can calculate magnitudes directly. In this case, the sGA is an IIR filter bank in which the window function is exponential like this, for each samples:

$\begin f3_(n) = n + \mathrm \cdot f1 - f2 \\ f2_(n) = f1 \cdot \mathrm \\ f1_(n) = f3 \cdot \mathrm \end$

where coeff is $2 \cos\left(\frac\right)$ and damping is $e^$. However, this implementation only covers first-order ones although it can be cascaded.

References

FFT algorithms

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