Implementation via STFTs for Audio (Fitz, Haken, and Christensen, ICMC 2000)

For some discrete time domain signal $x(n)$, equations 10 and 11 become

$$\hat{t}_{k,n} = t_n - \Re\left\{\ensuremath{\frac{X_{th;k,n} X^*_{h;k,n}}{\vert X_{h;k,n}\vert}}\right\}$$ (12)

$$\hat{\omega}_{k,n} = \omega_k +\Im\left\{\ensuremath{\frac{X_{dh;k,n} X^*_{h;k,n}}{\vert X_{h;k,n}\vert}}\right\}$$ (13)

Where the authors use a conventional DFT of a windowed signal:

$$X_{h;k,n} = \sum_{l=-\infty}^\infty h_{n-l} x_l exp(-j2\pi l k/N)$$ (14)

and also define time and frequency derivatives of the window and use them in ``modified'' DFTs:

$$X_{dh;k,n} = \sum_{l=-\infty}^\infty \ensuremath{\frac{d h_{n-l}}{dt}}x_l exp(-j2\pi l k/N)$$ (15)

$$X_{th;k,n} = \sum_{l=-\infty}^\infty t\cdot h_{n-l} x_l exp(-j2\pi l k/N)$$ (16)

(See overhead figures for illustrations of windows.)

NOTES:

• Application allows transients to be seen easily
• Errors still occur in data, but ``pruning'' possible.
• NEAT: shows that phase information CAN be used to get more meaningful parameters.