Overlap-Add Decomposition

Consider breaking an input signal $x$ into frames using a finite, zero-phase, length $M$ window $w$ . Then we may express the $m$ th windowed data frame as

$$x_m(n) \isdefs x(n)w(n-mR),\quad n \in (-\infty, +\infty)$$ (9.17)

or

$$x_m \isdefs x \cdot \hbox{\sc Shift}_{mR}(w)$$ (9.18)

where

$$\begin{eqnarray*} R &\isdef & \hbox{frame step (hop size)}\\ m &\isdef & \hbox{frame index} \end{eqnarray*}$$

The hop size is the number of samples between the begin-times of adjacent frames. Specifically, it is the number of samples by which we advance each successive window.

Figure 8.8 shows an input signal (top) and three successive windowed data frames using a length $M=128$ causal Hamming window and 50% overlap ($R=M/2=64$ ).

Figure 8.8: Input signal (top) and three successive windowed data frames.

For frame-by-frame spectral processing to work, we must be able to reconstruct $x$ from the individual overlapping frames, ideally by simply summing them in their original time positions. This can be written as

$$\begin{eqnarray*} x(n) &=& \sum_{m=-\infty}^{\infty} x_m(n) \\ &=& \sum_{m=-\infty}^{\infty} x(n) w(n-mR) \\ &=& x(n) \sum_{m=-\infty}^{\infty} w(n-mR) \end{eqnarray*}$$

Hence, $x= \sum_{m} x_m$ if and only if

$$\zbox {\sum_{m\in\mathbb{Z}} w(n-mR) = 1, \,\forall n\in\mathbb{Z}.}$$ (9.19)

This is the constant-overlap-add (COLA)^9.6 constraint for the FFT analysis window $w$ . It has also been called the partition of unity property.

Figure 8.9: Example of 50% overlap-add for the Bartlett (triangular) window

Figure 8.9 illustrates the appearance of 50% overlap-add for the Bartlett (triangular) window. The Bartlett window is clearly COLA for a wide variety of hop sizes, such as $M/2$ , $M/3$ , and so on, provided $M/k$ is an integer (otherwise the underlying continuous triangular window must be resampled). However, when using windows defined in a library, the COLA condition should be carefully checked. For example, the following Matlab/Octave script shows that there is a problem with the standard Hamming window:

M = 33;          % window length
R = (M-1)/2;     % hop size
N = 3*M;         % overlap-add span
w = hamming(M);  % window
z = zeros(N,1);  plot(z,'-k');  hold on;  s = z;
for so=0:R:N-M
  ndx = so+1:so+M;        % current window location
  s(ndx) = s(ndx) + w;    % window overlap-add
  wzp = z; wzp(ndx) = w;  % for plot only 
  plot(wzp,'--ok');       % plot just this window
end
plot(s,'ok');  hold off;  % plot window overlap-add

The figure produced by this matlab code is shown in Fig.8.10. As can be seen, the equal end-points sum to form an impulse in each frame of the overlap-add.

Figure 8.10: Overlap-add example for the default Hamming window in Matlab.

The Matlab window functions (such as hamming) have an optional second argument which can be either 'symmetric' (the default), or 'periodic'. The periodic case is equivalent to

w = hamming(M+1); % symmetric case
w = w(1:M);       % delete last sample for periodic case

The periodic variant solves the non-constant overlap-add problem for even $M$ and $R=M/2$ , but not for odd $M$ . The problem can be solved for odd $M$ and $R=(M-1)/2$ while retaining symmetry as follows:

w = hamming(M); % symmetric case
w(1) = w(1)/2;  % repair constant-overlap-add for R=(M-1)/2
w(M) = w(M)/2;

Since different window types may add or subtract 1 to/from $M$ internally, it is best to check the result using test code as above to make sure the window is COLA at the desired hop size. E.g., in Matlab:

• hamming(M) $\isdef$
  .54 - .46*cos(2*pi*(0:M-1)'/(M-1));
  gives constant overlap-add for $R=(M-1)/2$ , $(M-1)/4$ , etc., when endpoints are divided by 2 or one endpoint is zeroed
• hanning(M) $\isdef$
  .5*(1 - cos(2*pi*(1:M)'/(M+1)));
  does not give constant overlap-add for $R=(M-1)/2$ , but does for $R=(M+1)/2$

• blackman(M) $\isdef$
  .42 - .5*cos(2*pi*m)' + .08*cos(4*pi*m)';
  where m = (0:M-1)/(M-1), gives constant overlap-add for $R=(M-1)/3$ when $M$ is odd and $R$ is an integer, and $R=M/3$ when $M$ is even and $R$ is integer.

In summary, all windows obeying the constant-overlap-add constraint will yield perfect reconstruction of the original signal $x$ from the data frames $x_m = x\cdot\hbox{\sc Shift}_{mR}(w)$ by overlap-add (OLA). There is no constraint on window type, only that the window overlap-adds to a constant for the hop size used. In particular, $R=1$ always yields a constant overlap-add for any window function. We will learn later (§8.3.1) that there is also a simple frequency-domain test on the window transform for the constant overlap-add property.

To emphasize an earlier point, if simple time-invariant FIR filtering is being implemented, and we don't need to work with the intermediate STFT, it is most efficient to use the rectangular window with hop size $R=M$ , and to set $M=N-L+1$ , where $L$ is the length of the filter $h$ and $N$ is a convenient FFT size. The optimum $M$ for a given $L$ is an interesting exercise to work out.