### Objective:

In signal processing, synthesis is important in addition to analysis. This is especially true for the modification of data. For the Short-Time Fourier Transformation, the synthesis is often done using a simple overlap add (OLA), which is the sum of the outputs of the filter. Also, the output is re-weighted with the analysis window, such as occurs when using the phase vocoder. It is often presumed that with standard windows this will give satisfactory results.

Aside from Gabor frame theory, if the well-known construction of synthesis windows was possible, it would guarantee perfect reconstruction. However, this method is not used often in signal processing algorithms.

### Method:

In this project, we will systematically investigate if and for which parameters the respective OLA synthesis with the original window gives good reconstruction. We will compare it to the reconstruction with the dual window, introducing and motivating it as perfect reconstruction overlap add (PROLA). We will show that this method is always preferable to others and that it can be calculated very efficiently.

### Application:

This is currently being implemented in STx. There the phase vocoder will have the option to guarantee perfect reconstruction, either with dual or tight windows.

### Partners:

Department of Mathematics, University of Wisconsin-Eau Claire