通过离散傅里叶变换的间隔传播
Interval propagation through the discrete Fourier transform
摘要：
我们提出了一个算法，通过离散傅里叶变换的时间间隔的前向传播。当计算实值和复值序列的傅里叶变换的幅度时，该算法产生最佳可能的界限。我们表明，计算精确的范围内的振幅可以实现一个详尽的检查所有可能的角落的区间域。然而，由于角点的数量随着间隔的数量呈指数增加，这种方法对于大间隔信号是不可行的。我们提供了一个算法，不需要这样一个穷举搜索，并表明，最好的可能的界限，可以得到传播复杂的对，只有从凸包的端点在每一项的傅立叶级数。由于凸包总是紧内接在各自的严格的边界框区间运算产生的，我们得出的结论是，所获得的边界是保证包围真值。
英文摘要：
We present an algorithm for the forward propagation of intervals through the discrete Fourier transform. The algorithm yields best-possible bounds when computing the amplitude of the Fourier transform for real and complex valued sequences. We show that computing the exact bounds of the amplitude can be achieved with an exhaustive examination of all possible corners of the interval domain. However, because the number of corners increases exponentially with the number of intervals, such method is infeasible for large interval signals. We provide an algorithm that does not need such an exhaustive search, and show that the best possible bounds can be obtained propagating complex pairs only from the convex hull of endpoints at each term of the Fourier series. Because the convex hull is always tightly inscribed in the respective rigorous bounding box resulting from interval arithmetic, we conclude that the obtained bounds are guaranteed to enclose the true values.
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