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

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FBS Fixed Modifications

Consider applying a fixed (time-invariant) filter $ H(\omega_k)$ to each $ X_m(\omega_k)$ before resynthesizing the signal:

$\displaystyle Y_m(\omega_k) = X_m(\omega_k)H(\omega_k) $

where, $ H(\omega_k)$ is the sampled frequency response of a filter with impulse response

$\displaystyle h(n) = \frac{1}{N} \sum_{k=0}^{N-1} H(\omega_k) e^{j\omega_kn}, \quad n=0,\ldots,N-1 $

Let's examine the result this has on the signal in the time domain:

\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-n)] \\ &=& (x*[\tilde{w} \cdot h])(m) \\ \end{eqnarray*}

We see that the result is $ x$ convolved with a windowed version of the impulse response $ h$. This is in contrast to the OLA technique where the result gave us a windowed $ x$ filtered by $ h$ without the window having any effect on the filter, provided it obeys the COLA constraint and sufficient zero padding is used to avoid time aliasing.

In other words, FBS gives

$\displaystyle y = x * [\tilde{w} \cdot h] \leftrightarrow X \cdot [{\tilde W}\ast H] $

while OLA gives (for $ R=1$)

$\displaystyle y = x * [W(0)\cdot h] \leftrightarrow X \cdot [W(0)\cdot H] $

  • In FBS, the analysis window $ w$ smoothes the filter frequency response by time-limiting the corresponding impulse response.

  • In OLA, the analysis window can only affect scaling.

For these reasons, FFT implementations of FIR filters normally use the Overlap-Add method.

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Previous: STFT with Modifications
Next: Time Varying Modifications in FBS
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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