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Spectral Audio Signal Processing
    The Filter Bank Summation (FBS) Interpretation of the Short Time Fourier Transform (STFT)
       Dual Views of the Short Time Fourier Transform (STFT)
          Filter-Bank Summation (FBS) Interpretation of the STFT

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Filter-Bank Summation (FBS) Interpretation of the STFT

We can group the terms in the STFT definition differently to obtain the filter-bank interpretation:

$\displaystyle X_m(\omega_k)$ $\displaystyle =$ $\displaystyle \sum_{n=-\infty}^\infty \underbrace{[ x(n)e^{-j\omega_k n}]}_{x_k(n)} w(n-m)$  
  $\displaystyle =$ $\displaystyle \left[x_k \ast \hbox{\sc Flip}(w)\right](m) \protect$ (10.2)

As will be explained further below (and illustrated further in Figures 9.3, 9.4, and 9.5), under the filter-bank interpretation, the spectrum of $ x$ is first rotated along the unit circle in the $ z$ plane so as to shift frequency $ \omega_k$ down to 0 (via modulation by $ e^{-j\omega_k n}$ in the time domain), thus forming the heterodyned signal $ x_k(n)\isdeftext x(n)\exp(-j\omega_k n)$. Next, the heterodyned signal $ x_k(n)$ is lowpass-filtered to a narrow band about frequency 0 (via convolving with the time-reversed window $ \hbox{\sc Flip}(w)$). The STFT is thus interpreted as a frequency-ordered collection of narrow-band time-domain signals, as depicted in Fig.9.2. In other words, the STFT can be seen as a uniform filter bank in which the input signal $ x(n)$ is converted to a set of $ N$ time-domain output signals $ X_n(\omega_k)$, $ k=0,1,\ldots,N-1$, one for each channel of the $ N$-channel filter bank.

Figure 9.2: Filter Bank Summation (FBS) view of the STFT
\begin{figure}\input fig/fbs.pstex_t \end{figure}

Expanding on the previous paragraph, the STFT (9.2) is computed by the following operations:

  • Frequency-shift $ x(n)$ by $ -\omega_k$ to get $ x_k(n) \mathrel{\stackrel{\Delta}{=}}e^{-j\omega_k n}x(n)$.
  • Convolve $ x_k(n)$ with $ {\tilde w}\mathrel{\stackrel{\Delta}{=}}\hbox{\sc Flip}(w)$ to get $ X_m(\omega_k)$:

    $\displaystyle X_m(\omega_k) = \sum_{n=-\infty}^\infty x_k(n){\tilde w}(m-n) = (x_k * {\tilde w})(m) $

The STFT output signal $ X_m(\omega_k)$ is regarded as a time-domain signal (time index $ m$) coming out of the $ k$th channel of an $ N$-channel filter bank. The center frequency of the $ k$th channel filter is $ \omega_k = 2\pi k/N$, $ k=0,1,\ldots,N-1$. Each channel output signal is a baseband signal; that is, it is centered about dc, with the ``carrier term'' $ e^{j\omega_k m}$ taken out by ``demodulation'' (frequency-shifting). In particular, the $ k$th channel signal is constant whenever the input signal happens to be a sinusoid tuned to frequency $ \omega_k$ exactly.

Note that the STFT analysis window $ w$ is now interpreted as (the flip of) a lowpass-filter impulse response. Since the analysis window $ w$ in the STFT is typically symmetric, we usually have $ \hbox{\sc Flip}(w)=w$. This filter is effectively frequency-shifted to provide each channel bandpass filter. If the cut-off frequency of the window transform is $ \omega_c$ (typically half a main-lobe width), then each channel signal can be downsampled significantly. This downsampling factor is the FBS counterpart of the hop size $ R$ in the OLA context.

Figure 9.3 illustrates the filter-bank interpretation for $ R=1$ (the ``sliding STFT''). The input signal $ x(n)$ is frequency-shifted by a different amount for each channel and lowpass filtered by the (flipped) window.


\begin{psfrags} % latex2html id marker 24712\psfrag{w}{\Large$\protect\hbox{\s... ... where $x_k(n)\isdeftext x(n)\exp(-j\omega_k n)$. } \end{figure} \end{psfrags}

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Previous: Overlap-Add (OLA) Interpretation of the STFT
Next: FBS and Perfect Reconstruction
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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