Technical report: Oversampled windowed Fourier transform and filter banks
Oversampling enables redundant signal representations. In filter bank systems, a higher stopband attenuation, a better noise reduction, and more detailed subband signals can be achieved as compared to critically sampled filter banks. Oversampling in filter bank systems is further needed if subband signals are modified before they are used for signal synthesis. Any subband signal modification creates aliasing which can commonly be reduced by oversampling. However, oversampling increases the computational cost. Therefore, we investigate two classes of efficient signal processing systems which are well suited for oversampling purposes.
In the first part of this report, we revisit the well-established windowed Fourier transform as an important tool for signal analysis in the time-frequency domain. In addition, it is the basis of many signal processing applications when implemented as an oversampled analysis/synthesis system. A well-known example is the overlap-add FFT filter bank which is used in many advanced audio applications like signal enhancement, denoising, and signal compression. Normally, the overlap-add FFT filter bank is designed as a near perfect signal reconstruction system. We show that perfect reconstruction is possible by a proper design of the window function. In addition, this property is guaranteed for any fractional oversampling factor.
In the second part of this report, we present a fast design method for DFT filter banks with fractional oversampling, and for cosine modulated filter banks with integer oversampling . As compared to the overlap-add FFT filter bank, these systems split the input signal in a smaller number of subband signals. Our design technique is based on an iterated quadratic programming method developed by the author.
We illustrate our design methods with several representative examples. These examples and more
can be reproduced by a set of MATLAB programs available at the author’s home page.