Signal processing has entered into today's life on a broad range, from mobile phones, UMTS, xDSL, and digital television to scientific research such as psychoacoustic modeling, acoustic measurements, and hearing prosthesis. Such applications often use time-invariant filters by applying the Fourier transform to calculate the complex spectrum. The spectrum is then multiplied by a function, the so-called transfer function. Such an operator can therefore be called a Fourier multiplier. Real life signals are seldom found to be stationary. Quasi-stationarity and fast-time variance characterize the majority of speech signals, transients in music, or environmental sounds, and therefore imply the need for non-stationary system models. Considerable progress can be achieved by reaching beyond traditional Fourier techniques and improving current time-variant filter concepts through application of the basic mathematical concepts of frame multipliers.
Several transforms, such as the Gabor transform (the sampled version of the Short-Time Fourier Transformation), the wavelet transform, and the Bark, Mel, and Gamma tone filter banks are already in use in a large number of signal processing applications. Generalization of these techniques can be obtained via the mathematical frame theory. The advantage of introducing the frame theory consists particularly in the interpretability of filter and analysis coefficients in terms of frequency and time localization, as opposed to techniques based on orthonormal bases.
One possibility to construct time-variant filters exists through the use of Gabor multipliers. For these operators the result of a Gabor transform is multiplied by a given function, called the time-frequency mask or symbol, followed by re-synthesis. These operators are already used implicitly in engineering applications, and have been investigated as Gabor filters in the fields of mathematics and signal processing theory. If alternative transforms are used, the concept of multipliers can be extended appropriately. So, for example, the concept of wavelet multipliers could be investigated for a wavelet transform.
Different kinds of applications call for different frames. Multipliers can be generalized to the abstract level of frames without any further structure. This concept will be further investigated in this project. Its feasibility will be evaluated in acoustic applications using special cases of Gabor and wavelet systems.
The project goal is to study both the mathematical theory of frame multipliers and their application among selected problems in acoustics. The project is divided into the following subprojects: