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Spectral Audio Signal Processing
    The Filter Bank Summation (FBS) Interpretation of the Short Time Fourier Transform (STFT)
       STFT with Modifications
          Time Varying Modifications in FBS

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Time Varying Modifications in FBS

Consider now applying a time varying modification.

$\displaystyle Y_m(\omega_k) = X_m(\omega_k)H_m(\omega_k) \qquad \hbox{($R=1$)} $

where

$\displaystyle H_m(\omega_k) \leftrightarrow h_m(n) = \frac{1}{N} \sum_{k=0}^{N-1} H_m(\omega_k) e^{j\omega_kn} $

$ h_m(n)$ refers to the $ n^{th}$ tap of the FIR filter at time $ m$.

\begin{eqnarray*} y(m) &=& \frac{1}{N} \sum_{k=0}^{N-1} Y_m(\omega_k) e^{j\omega... ...tilde{w}(m-n)h_m(m-n)] \\ &=& (x*[\tilde{w} \cdot h_m])(m) \\ \end{eqnarray*}

Hence, the result is the convolution of $ x$ with the windowed version of $ h_m$.

  • We saw that in OLA with time varying modifications and $ R=1$ (a ``sliding'' DFT), the window served as a lowpass filter on each individual tap of the FIR filter being implemented.

  • In the more typical case in which $ R$ is the window length $ M$ divided a small integer like $ 2$-$ 10$, we may think of the window as specifying a type of cross-fade from the LTI filter for one frame to the LTI filter for the next frame.

  • Using a Bartlett (triangular) window with $ 50$% overlap, ($ R=2$), the sequence of FIR filters used is obtained simply by linearly interpolating the LTI filter for one frame to the LTI filter for the next.

  • In FBS, there is no limitation on how fast the filter $ h_m$ may vary with time, but its length is limited to that of the window $ w$.

  • In OLA, there is no limit on length (just add more zero-padding), but the filter taps are band-limited to the spectral width of the window.

  • FBS filters are time-limited by $ w$, while OLA filters are band-limited by $ w$ (another dual relation).

  • Recall for comparison that each frame in the OLA method is filtered according to

    $\displaystyle Y_m = X_m \cdot H_m = [X*W_m] \cdot H_m \leftrightarrow \underbrace{[x \cdot w_m]}_{x_m} * h_m $

    where $ w_m$ denotes $ \hbox{\sc Shift}_{mR}(w)$.
  • Time-varying FBS filters instantly in ``steady state''
  • FBS filters must be changed very slowly to avoid clicks and pops (discontinuity distortion likely when filter changes)
For more details, see [10].

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Previous: FBS Fixed Modifications
Next: STFT Summary and Conclusions
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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