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Downsampled STFT Filter Banks

We now look at STFT filter banks which are downsampled by the factor $ R>1$ . The downsampling factor $ R$ corresponds to a hop size of $ R$ samples in the overlap-add view of the STFT. From the filter-bank point of view, the impact of $ R>1$ is aliasing in the channel signals when the lowpass filter (analysis window) is less than ideal. When the conditions for perfect reconstruction are met, this aliasing will be canceled in the reconstruction (when the filter-bank channel signals are remodulated and summed).

Downsampled STFT Filter Bank

So far we have considered only $ R=1$ (the ``sliding'' DFT) in our filter-bank interpretation of the STFT. For $ R>1$ we obtain a downsampled version of $ X_m(\omega_k)$ :

\begin{eqnarray*} X_{mR}(\omega_k) &=& \sum_{n=-\infty}^\infty [x(n)e^{-j\omega_kn}]\tilde{w}(mR-n) \hspace{1.2cm} (\tilde{w} \mathrel{\stackrel{\Delta}{=}}\hbox{\sc Flip}(w)) \\ &=& (x_k \ast {\tilde w})(mR) \end{eqnarray*}

Let us define the downsampled time index as $ \tilde{m} \mathrel{\stackrel{\Delta}{=}}mR$ so that

$\displaystyle X_{\tilde{m}}(\omega_k) = \sum_{n=-\infty}^\infty [x(n)e^{-j\omega_kn}]\tilde{w}(\tilde{m}-n) \mathrel{\stackrel{\Delta}{=}}\left(x_k \ast {\tilde w}\right)(\tilde{m})$ (10.25)

i.e., $ X_{\tilde{m}}$ is simply $ X_m$ evaluated at every $ R^{th}$ sample, as shown in Fig.9.17.

\begin{psfrags} % latex2html id marker 25320\psfrag{w}{{\Large $\protect\hbox{\sc Flip}(w)$\ }}\psfrag{x(n)}{\Large $x(n)$\ }\psfrag{Xm}{\Large $X_m$\ }\psfrag{Xmt}{\Large $X_{\tilde{m}}$\ }\psfrag{X0}{\Large $X_{\tilde{m}}(\omega_0)$\ }\psfrag{X1}{\Large $X_{\tilde{m}}(\omega_1)$\ }\psfrag{XNm1}{\Large $X_{\tilde{m}}(\omega_{N-1})$\ }\psfrag{ejw0}{\Large $e^{-j\omega_0n}$\ }\psfrag{ejw1}{\Large $e^{-j\omega_1n}$\ }\psfrag{ejwNm1}{\Large $e^{-j\omega_{N-1}n}$\ }\psfrag{dR}{\Large $\downarrow R$\ }\begin{figure}[htbp] \includegraphics[width=\twidth]{eps/fbs2} \caption{Downsampled STFT filter bank.} \end{figure} \end{psfrags}

Note that this can be considered an implementation of a phase vocoder filter bank [212]. (See §G.5 for an introduction to the vocoder.)

Filter Bank Reconstruction

\begin{psfrags} % latex2html id marker 25351\psfrag{w}{{\Large $f$\ }} % should fix source (.draw file)\begin{figure}[htbp] \includegraphics[width=\twidth]{eps/FBSreconstruct} \caption{Interpolated, remodulated, filter-bank sum.} \end{figure} \end{psfrags}

Since the channel signals are downsampled, we generally need interpolation in the reconstruction. Figure 9.18 indicates how we might pursue this. From studying the overlap-add framework, we know that the inverse STFT is exact when the window $ w(n)$ is $ \hbox{\sc Cola}(R)$ , that is, when $ \hbox{\sc Alias}_R(w)$ is constant. In only these cases can the STFT be considered a perfect reconstruction filter bank. From the Poisson Summation Formula in §8.3.1, we know that a condition equivalent to the COLA condition is that the window transform $ W(\omega)$ have notches at all harmonics of the frame rate, i.e., $ W(2\pi k/R)=0$ for $ k=1,2,3,R-1$ . In the present context (filter-bank point of view), perfect reconstruction appears impossible for $ R>1$ , because for ideal reconstruction after downsampling, the channel anti-aliasing filter ($ w$ ) and interpolation filter ($ f$ ) have to be ideal lowpass filters. This is a true conclusion in any single channel, but not for the filter bank as a whole. We know, for example, from the overlap-add interpretation of the STFT that perfect reconstruction occurs for hop-sizes greater than 1 as long as the COLA condition is met. This is an interesting paradox to which we will return shortly.

What we would expect in the filter-bank context is that the reconstruction can be made arbitrarily accurate given better and better lowpass filters $ w$ and $ f$ which cut off at $ \omega_c = \pi/R$ (the folding frequency associated with down-sampling by $ R$ ). This is the right way to think about the STFT when spectral modifications are involved.

In Chapter 11 we will develop the general topic of perfect reconstruction filter banks, and derive various STFT processors as special cases.

Downsampling with Anti-Aliasing

Figure 9.19: Processing in one filter-bank analysis channel.
\includegraphics[width=3in]{eps/FBSonechan}

In OLA, the hop size $ R$ is governed by the COLA constraint

$\displaystyle \sum_{m=-\infty}^\infty w(n+mR) = \hbox{constant}$ (10.26)

In FBS, $ R$ is the downsampling factor in each of the filter-bank channels, and thus the window $ w$ serves as the anti-aliasing filter (see Fig.9.19). We see that to avoid aliasing, $ W(\omega)$ must be bandlimited to $ (-\pi/R, \pi/R)$ , as illustrated schematically in Fig.9.20.

Figure 9.20: Schematic illustration of a window transform that suppresses all aliasing.
\includegraphics[width=3in]{eps/IdealWindowFD}

Properly Anti-Aliasing Window Transforms

For simplicity, define window-transform bandlimits at first zero-crossings about the main lobe. Given the first zero of $ W(\omega)$ at $ L \frac{2\pi}{M} \leq \frac{\pi}{R}$ , we obtain

$\displaystyle \zbox {R_{\hbox{max}}= \frac{M}{2L}}$ (10.27)

The following table gives maximum hop sizes for various window types in the Blackman-Harris family, where $ L$ is both the number of constant-plus-cosine terms in the window definition (§3.3) and the half-main-lobe width in units of side-lobe widths $ 2\pi/M$ . Also shown in the table is the maximum COLA hop size we determined in Chapter 8.
L Window Type (Length $ M$ ) $ R_{\hbox{max}}$ $ R_{\hbox{\hbox{\sc Cola}}}$
1 Rectangular M/2 M
2 Generalized Hamming M/4 M/2
3 Blackman Family M/6 M/3
L $ L$ -term Blackman-Harris M/2L M/L
In the table, any $ R\leq R_{\hbox{max}}$ suppresses aliasing well.

It is interesting to note that the maximum COLA hop size is double the maximum downsampling factor which avoids aliasing of the main lobe of the window transform in FFT-bin signals $ X_{\tilde m}(\omega_k)$ . Since the COLA constraint is a sufficient condition for perfect reconstruction, this aliasing is quite heavy (see Fig.9.21), yet it is all canceled in the reconstruction. The general theory of aliasing cancellation in perfect reconstruction filter banks will be taken up in Chapter 11.

Figure 9.21: Illustration of main-lobe aliasing intervals.
\includegraphics[width=3in]{eps/WindowAliasingFD}

It is important to realize that aliasing cancellation is disturbed by FBS spectral modifications.10.4For robustness in the presence of spectral modifications, it is advisable to keep $ R\leq R_{\hbox{max}}= M/(2L)$ . For compression, it is common to use $ R = 2 R_{\hbox{max}}= R_{\hbox{\hbox{\sc Cola}}} = M/L$ together with a ``synthesis window'' in a weighted overlap-add (WOLA) scheme (§8.6).

Hop Sizes for WOLA

In the weighted overlap-add method, with the synthesis (output) window equal to the analysis (input) window, we have the following modification of the recommended maximum hop-size table:

L In and Out Window (Length $ M$ ) $ R_{\hbox{max}}$ $ R_{\hbox{\hbox{\sc Cola}}}$
1 Rectangular ($ L=1$ ) M/2 M
2 Generalized Hamming ($ L=2$ ) M/6 M/3
3 Blackman Family ($ L=3$ ) M/10 M/5
L $ L$ -term Blackman-Harris M/(4L-2) M/(2L-1)
Note that the following properties hold as before in the OLA case:
  • $ R_{\hbox{max}}$ is equal to $ 2\pi$ divided by the main-lobe width in ``side lobes'', while

  • $ R_{\hbox{\hbox{\sc Cola}}}$ is $ 2\pi$ divided by the first notch frequency in the window transform (lowest available frame rate at which all frame-rate harmonics are notched).

  • For windows in the Blackman-Harris families, and with main-lobe widths defined from zero-crossing to zero-crossing, $ R_{\hbox{\hbox{\sc Cola}}} = 2 R_{\hbox{max}}$ .

Constant-Overlap-Add (COLA) Cases

  • Weak COLA: Window transform has zeros at frame-rate harmonics:

    $\displaystyle W(\omega_k) = 0, \quad k = 1,2, \dots, R-1, \quad \omega_k \isdef \frac{2\pi k}{R} $

  • Strong COLA: Window transform is bandlimited consistent with downsampling by the frame rate:

    $\displaystyle W(\omega) = 0, \quad \vert\omega\vert \geq \pi/R $

    • Perfect OLA reconstruction
    • No aliasing
    • better for spectral modifications
    • Time-domain window infinitely long in ideal case

Hamming Overlap-Add Example

Matlab code: 

M = 33;         % window length
w = hamming(M);
R = (M-1)/2;    % maximum hop size
w(M) = 0;       % 'periodic Hamming' (for COLA)
%w(M) = w(M)/2; % another solution,
%w(1) = w(1)/2; %  interesting to compare

Figure 9.22: Periodic-Hamming OLA waveforms.
\includegraphics[width=\textwidth ]{eps/olaHammingC}

Periodic-Hamming OLA from Poisson Summation Formula

Matlab code: 

ff = 1/R; % frame rate (fs=1)
N = 6*M;  % no. samples to look at OLA
sp = ones(N,1)*sum(w)/R; % dc term (COLA term)
ubound = sp(1);  % try easy-to-compute upper bound
lbound = ubound; % and lower bound
n = (0:N-1)';
for (k=1:R-1) % traverse frame-rate harmonics
  f=ff*k;
  csin = exp(j*2*pi*f*n); % frame-rate harmonic
  % find exact window transform at frequency f
  Wf = w' * conj(csin(1:M));
  hum = Wf*csin;   % contribution to OLA "hum"
  sp = sp + hum/R; % "Poisson summation" into OLA
  % Update lower and upper bounds:
  Wfb = abs(Wf);
  ubound = ubound + Wfb/R; % build upper bound
  lbound = lbound - Wfb/R; % build lower bound
end

In this example, the overlap-add is theoretically a perfect constant (equal to $ 1.08$ ) because the frame rate and all its harmonics coincide with nulls in the window transform (see Fig.9.24). A plot of the steady-state overlap-add and that computed using the Poisson Summation Formula (not shown) is constant to within numerical precision. The difference between the actual overlap-add and that computed using the PSF is shown in Fig.9.23. We verify that the difference is on the order of $ 10^{-15}$ , which is close enough to zero in double-precision (64-bit) floating-point computations. We thus verify that the overlap-add of a length $ 33$ Hamming window using a hop size of $ R = (M-1)/2 = 16$ samples is constant to within machine precision.

Figure 9.23: Periodic-Hamming Poisson summation formula test.
\includegraphics[width=\textwidth ]{eps/olassmmpHammingC}

Figure 9.24 shows the zero-padded DFT of the modified Hamming window we're using ( $ w(M)\leftarrow 0$ ) with the frame-rate harmonics marked. In this example ($ R=M/2$ ), the upper half of the main lobe aliases into the lower half of the main lobe. (In fact, all energy above the folding frequency $ \pi/R$ aliases into the lower half of the main lobe.) While this window and hop size still give perfect reconstruction under the STFT, spectral modifications will disturb the aliasing cancellation during reconstruction. This ``undersampled'' configuration is suitable as a basis for compression applications.

Figure 9.24: Hamming window transform and frame-rate.
\includegraphics[width=\textwidth ]{eps/windowTransformHammingC}

Note that if we were to cut $ R$ in half to $ R=M/4$ , then the folding frequency in Fig.9.24 would coincide with the first null in the window transform. Since the frame rate and all its harmonics continue to land on nulls in the window transform, overlap-add is still exact. At this reduced hop size, however, the STFT becomes much more robust to spectral modifications, because all aliasing in the effective downsampled filter bank is now weighted by the side lobes of the window transform, with no aliasing components coming from within the main lobe. This is the central result of [9].

Kaiser Overlap-Add Example

Matlab code: 

M = 33;    % Window length
beta = 8;
w = kaiser(M,beta);
R = floor(1.7*(M-1)/(beta+1)); % ROUGH estimate (gives R=6)

Figure 9.25 plots the overlap-added Kaiser windows, and Fig.9.26 shows the steady-state overlap-add (a time segment sometime after the first 30 samples). The ``predicted'' OLA is computed using the Poisson Summation Formula using the same matlab code as before. Note that the Poisson summation formula gives exact results to within numerical precision. The upper (lower) bound was computed by summing (subtracting) the window-transform magnitudes at all frame-rate harmonics to (from) the dc gain of the window. This is one example of how the PSF can be used to estimate upper and lower bounds on OLA error.

Figure 9.25: Kaiser OLA waveforms.
\includegraphics[width=\textwidth ]{eps/olakaiserC}

Figure 9.26: Kaiser OLA, steady state.
\includegraphics[width=\textwidth ]{eps/olasskaiserC}

The difference between measured steady-state overlap-add and that computed using the Poisson summation formula is shown in Fig.9.27. Again the two methods agree to within numerical precision.

Figure 9.27: Kaiser OLA from Poisson summation formula minus computed OLA.
\includegraphics[width=\textwidth ]{eps/olassmmpkaiserC}

Finally, Fig.9.28 shows the Kaiser window transform, with marks indicating the folding frequency at the chosen hop size $ R$ , as well as the frame-rate and twice the frame rate. We see that the frame rate (hop size) has been well chosen for this window, as the folding frequency lies very close to what would be called the ``stop band'' of the Kaiser window transform. The ``stop-band rejection'' can be seen to be approximately $ 58$ dB (height of highest side lobe in Fig.9.28). We conclude that this example--a length 33 Kaiser window with $ \beta=8$ and hop-size $ R=6$ -- represents a reasonably high-quality audio STFT that will be robust in the presence of spectral modifications. We expect such robustness whenever the folding frequency lies above the main lobe of the window transform.

Figure 9.28: Kaiser window transform and frame-rate.
\includegraphics[width=\textwidth ]{eps/windowTransformkaiserC}

Remember that, for robustness in the presence of spectral modifications, the frame rate should be more than twice the highest main-lobe frequency.

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STFT with Modifications
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Portnoff Windows