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Group theoretical approach to Gabor analysis

机译：小组理论方法进行Gabor分析

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We describe new methods to obtain nonorthogonal Gabor expansions of discrete and finite signals and reconstruction of signals from regularly sampled short time Fourier transform (STFT) values by series expansions. By this we understand the expansion of a signal of a given length n into a (finite) series of coherent building blocks, obtained from a Gabor atom through discrete time- and frequency-shift operators. Although bump-type atoms are natural candidates, the approach is not restricted to such building blocks. Also the set of time- and frequency-shift operators does not have to be a (product) lattice, but just an ordinary (additive) subgroup of the time/frequency plane, which is naturally identified with the 2-D n x n cyclic group. In contrast, other nonseparable subgroups turn out to be more interesting for our task: the efficient determination of a suitable set of coefficients for the coherent expansion. It is sufficient to determine the so-called dual Gabor atom. The existence and basic properties of this dual atom are well known in the case of lattice groups. It is shown that this is true for general groups. But more importantly, we demonstrate that the conjugate gradient methpd reduces the computational complexity drastically.
机译：我们描述了获取离散和有限信号的非正交Gabor展开以及通过级数展开从常规采样的短时傅立叶变换（STFT）值重构信号的新方法。通过这种方式，我们可以理解将给定长度n的信号扩展为（有限的）一系列相干构件，这些构件是通过离散的时移和频移算符从Gabor原子获得的。尽管凸点型原子是自然的候选者，但该方法并不限于此类构造块。同样，时移和频移算子集也不必是（乘积）晶格，而只是时间/频率平面的普通（加法）子组，它自然地由2-D n x n循环组标识。相反，对于我们的任务，其他不可分割的子组变得更加有趣：有效确定一组合适的系数用于相干扩展。确定所谓的双重Gabor原子就足够了。在晶格基团的情况下，该双原子的存在和基本性质是众所周知的。事实证明，这对于一般人群是正确的。但更重要的是，我们证明了共轭梯度方法极大地降低了计算复杂度。

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