To ensure successful reconstruction of nonmodified spectra, the analysis
window must satisfy the COLA constraint. In general, if the analysis window satisfies
the condition $$\sum _{m=-\infty}^{\infty}{g}^{a+1}(n-mR)=c\text{\hspace{0.17em}}}\forall n\in \mathbb{Z$$, the window is considered to be COLA-compliant. Additionally, COLA
compliance can be described as either weak or strong.

Weak COLA compliance implies that the Fourier transform of the analysis
window has zeros at frame-rate harmonics such that

Alias cancellation is disturbed by spectral modifications. Weak COLA
relies on alias cancellation in the frequency domain. Therefore, perfect
reconstruction is possible using weakly COLA-compliant windows as long as
the signal has not undergone any spectral modifications.

For strong COLA compliance, the Fourier transform of the window must be
bandlimited consistently with downsampling by the frame rate such that

This equation shows that no aliasing is allowed by the
strong COLA constraint. Additionally, for strong COLA compliance, the value
of the constant $$c$$ must equal 1. In general, if the short-time spectrum is
modified in any way, a stronger COLA compliant window is preferred.

You can use the `iscola`

function to check for weak COLA compliance. The
number of summations used to check COLA compliance is dictated by the window length and
hop size. In general, it is common to use $$a=1$$ in $$\sum _{m=-\infty}^{\infty}{g}^{a+1}(n-mR)=c\text{\hspace{0.17em}}}\forall n\in \mathbb{Z$$ for weighted overlap-add (WOLA), and $$a=0$$ for overlap-add (OLA). By default, `istft`

uses the
WOLA method, by applying a *synthesis window* before performing the
overlap-add method.

In general, the synthesis window is the same as the analysis window. You can construct
useful WOLA windows by taking the square root of a strong OLA window. You can use this
method for all nonnegative OLA windows. For example, the root-Hann window is a good
example of a WOLA window.