Overlap & Reconstruction
We solved the FFT boundary problem by fading the edges of each block. Great. But now we've created another problem: we faded the edges of each block. Luckily, audio engineers already know the basic idea behind the solution.
1 · You already know this: the crossfade
Put two edits hard against each other and a discontinuity can click. In a DAW, we overlap the regions and fade one down while the next fades up.
2 · Now make the regions FFT blocks
Each analysis frame begins at a new position called the hop. If the hop is smaller than the block, the blocks overlap.
3 · Slide the windows across each other
Each curve is one neighbouring block's weighting function. The block size stays the same; changing overlap changes how far apart their starting positions are.
4 · ADD the overlapping contributions
Now look underneath the individual windows. Each coloured strip is a neighbouring block. Its colour fades with the window weighting, so overlapping blocks blend into one another rather than looking like hard-butted regions.
A good overlap/window combination keeps the reconstruction continuously covered. Poor combinations reveal pale gaps or uneven bands.
MATHEMATICAL CHECK · sum of overlapping window weights
5 · Why hop size is different from FFT size
how long each analysis looks
↓
bin spacing / observation duration
how far we move before the next analysis
↓
frame/update spacing
At 48 kHz, a 1024-sample FFT observes about 21.33 ms. With 50% overlap, the next frame starts only 512 samples — 10.67 ms — later.
This is why a spectral process can use a relatively long FFT while still producing analysis frames more frequently than once per whole block.
Hop size is like a frame rate
If you have worked with video, motion capture or frame-based analysis, hop size is an intuitive idea. The FFT analyses one block of audio, then moves forward by the hop size and analyses again.
A large hop means fewer spectral snapshots per second. A small hop means more frequent snapshots — and therefore more overlap between neighbouring blocks.
For example, at 48 kHz with a hop of 512 samples:
The important distinction is that FFT size determines how much audio is contained in each analysis frame, while hop size determines how often a new frame begins.
Because the frames can overlap, a new analysis can begin before the previous block has finished.
6 · The important technical wrinkle
For actual FFT processing, perfect reconstruction depends on the complete analysis window + synthesis window + hop scheme. Some systems window before the FFT and again after the IFFT; in that case it may be the combined weighting that must satisfy the reconstruction condition.
The general family of conditions is often discussed using COLA — Constant OverLap-Add. The exact valid combinations depend on the window definition and processing method.
7 · From analyser to processor
Up to now we've mostly looked at spectra. But if we can transform a block into bins, change those bins, transform it back and reconstruct the overlapping blocks...
Draw on the Spectrum · Spectral EQ
Take the magnitude of individual frequency bins and multiply them by a gain curve.
Our old friend MULTIPLY is about to become a drawable spectral filter.