多窗实值离散Gabor变换及其快速算法

发布时间：2018-04-03 04:18

  本文选题：实值离散Gabor变换　切入点：双正交性　出处：《安徽大学》2016年博士论文

【摘要】：Gabor变换是重要的时频分析方法之一,广泛应用于非平稳信号的检测、分析与处理。然而由于Heisenberg不确定原理的制约,传统的单窗Gabor变换的时频局域性(或者时频分辨精度)受到很大限制。由于分析窗和综合窗函数宽度是固定的,由单窗Gabor变换获得的时频谱时间分辨精度与频率分辨精度也是固定的,并且不可能同时都好,二者是矛盾的。采用宽窗将产生高频率分辨率但低时间分辨率的时频谱；反之,使用窄窗将产生高时间分辨率但低频率分辨率的时频谱。于是学者们在框架理论基础上提出了多窗复值离散Gabor变换(M-CDGT),通过组合多窗情况下获得的时频谱,可有效改善时频表示精度。但M-CDGT的研究还不够完善和深入,为了拓展多窗离散Gabor变换的研究,本文利用双正交分析法研究了多窗实值离散Gabor变换(M-RDGT)。主要研究内容和创新成果如下：提出了有限长序列M-RDGT及其快速算法。将周期的M-CDGT变换核中的复指数改换为离散Hartley变换(DHT)中的变换核cas实函数,从而提出了周期(有限长序列)的M-RDGT;利用双正交分析法研究了M-RDGT与其逆变换(即多窗实值离散Gabor展开)中的窗函数双正交关系式,并证明此关系式等同于M-RDGT的完备性条件；研究了利用窗函数双正交关系式求解窗函数的快速算法；研究了基于DHT的M-RDGT及其逆变换的快速算法；由于M-RDGT系数与M-CDGT系数之间关系就如同离散Fourier变换(DFT)系数与DHT系数之间关系(非常简单的代数关系),因此M-RDGT快速算法也提供了一种快速有效地计算M-CDGT方法。提出了超长(或无限长)序列M-RDGT及其快速算法。在有限长系列(周期)的M-RDGT中,窗函数长度与待分析序列长度必须相同,对于超长序列的M-RDGT计算,无疑将大幅增加求解窗函数所需的计算量及存储空间,有时甚至导致求解数值不稳定。为了使得窗函数长度不随待分析序列长度变化,即用长度较短的窗函数分析超长甚至无限长待分析序列,本文在有限长序列M-RDGT基础上研究了超长序列M-RDGT及其快速算法,推导了超长序列M-RDGT在满足完备性条件下新的窗函数双正交关系式。提出了多抽样率快速并行实现多窗实值离散Gabor变换方法。借助于多抽样率数字滤波器组的分析与综合基本原理和多窗离散Gabor展开与变换中分析(求变换系数)与综合(展开即信号重建)原理的相似性,设计了一种并行多抽样率分析与综合卷积器组来实现多窗实值离散Gabor展开与变换。所设计的分析和综合卷积组中的每一并行通道具有一致的结构并能够利用快速DHT算法减小计算量。每一并行通道计算复杂性只决定于输入离散信号的长度及Gabor频率抽样点数,不会随M-RDGT过抽样率及窗数增加而增大,因此,每一并行通道的计算复杂性非常小。最后,本文对包含冲激函数的正弦函数序列、指数衰减正弦类瞬变序列以及Apnea-ECG数据库中的心电(ECG)序列进行了M-RDGT和时频谱计算实验,实验结果表明所提出的M-RDGT提供了一种快速有效的方法分析和展示包含有多个或时变频率分量信号的动态时频内容。
[Abstract]:Gabor transform is a time-frequency analysis method is one of the important and widely used in the detection of non-stationary signal analysis and processing. However, due to the Heisenberg uncertainty principle constraints, the traditional single window Gabor transform time-frequency locality (or time-frequency resolution) is very limited. The width of window and window function comprehensive analysis is fixed, obtained by the single window Gabor transform spectrum time resolution and frequency resolution is fixed, and at the same time can not be all good, the two is contradictory. The wide window will produce high frequency resolution but low time resolution spectrum; on the other hand, the use of narrow window will have a high time resolution but when the spectrum of low frequency resolution. So the scholars in the theoretical framework is proposed based on the multi window complex valued discrete Gabor transform (M-CDGT), when the spectrum was obtained through the combination of multi window case, can effectively improve the time-frequency representation of the fine The degree of M-CDGT. But the research is still not perfect and thorough, in order to expand the research of multi window discrete Gabor transform, this paper studied the biorthogonal multi window real valued discrete Gabor transform (M-RDGT) method. The main research contents and innovations are as follows: the finite length sequence of M-RDGT and its fast algorithm. The cycle of the M-CDGT kernel the complex index change for the discrete Hartley transform (DHT) transform kernel CAS real function, thus put forward the cycle (finite length sequence) of M-RDGT; on M-RDGT and inverse analysis method using two orthogonal (i.e. multi window real valued discrete Gabor expansion) biorthogonal window function in the relationship, and to prove the completeness this relation is equivalent to M-RDGT; was studied by using the fast algorithm of Bi orthogonal relation window function solving window function; Study on fast algorithm of M-RDGT and the DHT inverse transformation based on M-RDGT and M-C; the coefficient The relationship between DGT coefficients as discrete Fourier transform (DFT) the relationship between the coefficient and DHT coefficient (algebraic relations very simple), so M-RDGT algorithm also provides a fast and effective method of calculating M-CDGT. Put forward long (or infinite) sequence of M-RDGT and the fast algorithm. In the limited series (long period) in M-RDGT, the window function and the length of stay of the length of the sequence must be the same, for long sequences of the M-RDGT calculation, will undoubtedly increase the amount of computation and storage space required for window function, sometimes even lead to numerical instability. In order to make the window length with sequence length changes to be analyzed, using window function analysis of long short length or even infinite long sequence analysis, this study of the long M-RDGT sequence and its fast algorithm in a finite length sequence of M-RDGT based on the deduced sequence of M-RDGT in length to meet the completeness Bi orthogonal relation window function under the new conditions. Proposed multirate fast parallel implementation of multi window real valued discrete Gabor transform method. With the analysis of analysis of multirate digital filter banks and the basic principle of comprehensive and multi window discrete Gabor expansion and transform (for transform coefficients) and synthesis (namely signal reconstruction) the similarity principle, a parallel multi rate sampling analysis and comprehensive convolver group to achieve multi window real valued discrete Gabor expansion and transform design. Each parallel channel analysis and comprehensive design of convolution group has the same structure and can use the fast DHT algorithm to reduce the computation of each parallel channel is calculated. The complexity depends only on the input length and Gabor frequency sampling points of discrete signals, not with the M-RDGT sampling rate and window number increases, therefore, the computational complexity of each parallel channel is very small. Finally, the The sequence contains sine impulse function, exponential sinusoidal transient sequences in the Apnea-ECG database and ECG (ECG) sequence of M-RDGT and spectrum calculation experiment, experimental results show that the proposed M-RDGT provides a fast and effective method of analysis and display with the dynamic variable frequency component signal multiple or when the frequency content.

【学位授予单位】：安徽大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：TN911.6
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本文编号：1703591