Filter-Bank Summation (FBS) Interpretation of the STFT

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

$$X_m(\omega_k) = \sum_{n=-\infty}^\infty \underbrace{[ x(n)e^{-j\omega_k n}]}_{x_k(n)} w(n-m)$$

$$= \left[x_k \ast \hbox{\sc Flip}(w)\right](m) \protect$$ (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

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 $X_m(\omega_k)$ :

  $$X_m(\omega_k) = \sum_{n=-\infty}^\infty x_k(n){\tilde w}(m-n) = (x_k * {\tilde w})(m)$$ (10.3)

  
  

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.