Discrete Inverse STFT

Gabor frame Inversion

We do not have in general: $$ x[t] = \sum_{\tau} \sum_{\nu} X[\tau,\nu] w[t-K\tau]e^{i2\pi\frac{\nu}{M}t} $$ With matrix notation: $$ \boldsymbol{x} \neq \boldsymbol{\Phi}\boldsymbol{\Phi^*} \boldsymbol{x} $$

The invert of a Gabor dictionary $\boldsymbol{\Phi}$ is obtained by the canonical dual $\boldsymbol{\tilde \Phi}$, which is also a Gabor transform constructed using a dual window $\boldsymbol{\tilde w}$:

$$ x[t] = \sum_{\tau} \sum_{\nu} X[\tau,\nu] \tilde w[t-K\tau]e^{i2\pi\frac{\nu}{M}t} $$ With matrix notation: $$ \boldsymbol{x} = \boldsymbol{\tilde \Phi} \boldsymbol{X} = \boldsymbol{\tilde \Phi} \boldsymbol{\Phi^*} \boldsymbol{x} $$

If the Gabor dictionary is a Parseval Frame (or a normalized tight frame), then $\boldsymbol{\tilde \Phi}= \boldsymbol{ \Phi}$, and $$ \boldsymbol{x} = \boldsymbol{ \Phi}\boldsymbol{X}= \boldsymbol{ \Phi}\boldsymbol{\Phi^*} \boldsymbol{x} $$ and $$ \Vert\boldsymbol{x}\Vert^2 = \Vert \boldsymbol{X}\Vert^2 $$