# Advantage of applying a Window Function in Analysis and Synthesis of STFT?

I'm studying Short-time Fourier Transform and I learned that if you apply a windowing other than rectangular, you need to overlap the frames---for example, 50% overlap for Hamming family---to reconstruct the signal properly (like add them up to 1). However, I only need to apply this window function during the Analysis part and I can reconstruct the signal properly during synthesis.

But in some literature (like Audacity's Noise Reduction) I saw some methods that apply a second window function during the synthesis part and employ a 75% overlap-add. What's the advantage of doing the window function twice (analysis/synthesis) with 75% overlap compared to a single window function (analysis) with 50% overlap? I can't seem to find any explanation justifying the advantage between these two windowing techniques.

Edit: I also tried to implement the 75% overlap with Hann window applied on both analysis and synthesis part and I was able to yield a x1.5 of the original signal's amplitude. Did I do it correctly?

The STFT is an instance of an Analysis/Synthesis system. If we restrict to windows of finite length, it amounts to windowing blocks of length $$L$$, taking the DFT, and hopping by $$K$$ samples to perform the same operation. An A/S system is said invertible, or perfect, if you can recover the original signal from the blocks of DFT coefficients.

If $$K>L$$, some samples are missed, there are holes between blocks, and generally some information is lost (unless you have additional information).

If $$K=L$$, blocks are contiguous, and do not overlap. Here, the transformation is critical, or non-redundant: you have the same numbers of Fourier coefficients than of samples. In that case, you can invert it even with non-rectangular windows. It suffices that the windows don't vanish, because windows without zero-values can be inverted (and Fourier as well). There is only one inverse in that case. This is for your first question.

If you allow overlap, then several things may happen. I shall come back later, but basically:

• a proper synthesis window can made a system with a vanishing analysis window invertible
• it can reduced artifacts caused when you perform processing after the analysis stage (frequency removal, etc.)

[Be back later]]

• Thank you for your initial insight. I actually haven't thought of the case where $K>L$ since it's apparent that we lose information if we do that, so I only studied/implemented three cases where I apply 1) $K=L$ hops and 2-3) $K<L$ hops (i.e. 50%, 75% overlap). So I would like to know more about the $K<L$ case and the significance of why do we need to apply a second window to the synthesis as well. – micropyre Feb 24 at 2:36
• I'll try to be back in the week-end – Laurent Duval Feb 25 at 23:59