The fractional Fourier transform (FrFT) is a generalization of the usual Fourier transform. The aim of this paper is to show the compression of sound signal in FrFT domain and to prove the qualitative and quantitative uncertainty principles for the FrFT. The first of these results consists the Hardy’s and an \(L^p-L^q\) version of Miyachi’s theorems for the FrFT, which estimates of decay of two fractional Fourier transforms \(F_{\alpha }(f)\) and \(F_{\gamma } (f)\), with \(\gamma -\alpha \ne n\pi , \forall n\in \mathbb {Z}\). The second result consists an extension of Faris’s local uncertainty principle which states that if a non zero function \(F_{\alpha }(f) \in L^2(\mathbb {R})\) is highly localized near a single point then \(F_{\gamma } (f)\) cannot be concentrated in a set of finite measure with \(\gamma -\alpha \ne n\pi , \forall n\in \mathbb {Z}\). From our results we deduce the usual uncertainty principles for the fractional Fourier transform which states these theorems between a function f and its fractional Fourier transform \(F_{\gamma }(f)\).

分数阶傅立叶变换（FrFT）是常规傅立叶变换的概括。本文的目的是展示FrFT域中声音信号的压缩，并证明FrFT的定性和定量不确定性原理。这些结果中的第一个包括Hardy以及FrFT的Miyachi定理的\（L ^ pL ^ q \）版本，该版本估计两个分数阶Fourier变换\（F _ {\ alpha}（f）\）和\ （F _ {\ gamma}（f）\），其中\（\ gamma-\ alpha \ ne n \ pi，\ forall n \ in \ mathbb {Z} \）。第二个结果包括Faris局部不确定性原理的扩展，该原理指出，如果非零函数\（F _ {\ alpha}（f）\ in L ^ 2（\ mathbb {R}）\）高度集中在单个点附近，则\（F _ {\ gamma}（f）\）不能集中在\（\ gamma-\ alpha \ ne n \ pi，\ forall n \ in \ mathbb中的一组有限度量中{Z} \）。从我们的结果中，我们得出分数傅里叶变换的通常不确定性原理，其中陈述了函数f及其分数傅里叶变换\（F _ {\ gamma}（f）\）之间的这些定理。