The spectrum of the convolution of two continuous functions can be determined as the continuous Fourier transform of the cross-correlation function. The same can be said about the spectrum of the convolution of two infinite discrete sequences, which can be determined as the discrete time Fourier transform of the cross-correlation function of the two sequences. In current digital signal processing, the spectrum of the contiuous Fourier transform and the discrete time Fourier transform are approximately determined by numerical integration or by densely taking the discrete Fourier transform. It has been shown that all three transforms share many analogous properties. In this paper we will show another useful property of determining the spectrum terms of the convolution of two finite length sequences by determining the discrete Fourier transform of the modified cross-correlation function. In addition, two properties of the magnitude terms of orthogonal wavelet scaling functions are developed. These properties are used as constraints for an exhaustive search to determine an robust lower bound on conjoint localization of orthogonal scaling functions.

中文翻译：

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正交标度函数的幅度项的性质。

两个连续函数的卷积谱可以确定为互相关函数的连续傅立叶变换。两个无限离散序列的卷积谱也可以这样说，可以确定为两个序列的互相关函数的离散时间傅立叶变换。在当前的数字信号处理中，连续傅立叶变换和离散时间傅立叶变换的频谱大致是通过数值积分或密集取离散傅立叶变换来确定的。已经表明，所有三个变换共享许多类似的特性。在本文中，我们将通过确定修正互相关函数的离散傅立叶变换来展示确定两个有限长度序列卷积的频谱项的另一个有用特性。此外，还开发了正交小波标度函数幅度项的两个性质。这些属性用作穷举搜索的约束，以确定正交标度函数的联合定位的稳健下界。