【摘要】：傅里葉分析在信號分析處理中做出了杰出的貢獻(xiàn)，但無論是在時域或者在頻域，它都是定義在整個域上的，它不能分析出某段時間內(nèi)某個頻段的信號特征，，即，傅里葉變換沒有時頻的局部化性能。小波分析的出現(xiàn)為信號處理領(lǐng)域提供了一種自適應(yīng)性的將時域和頻域同時局部化的分析方法，無論是分析低頻信號，還是分析高頻信號，它都能自動調(diào)節(jié)時頻窗的大小和形狀，以適應(yīng)實際分析的需要。同時，小波分析的多分辨率分析思想也給信號處理領(lǐng)域帶來了新的思路。Mallat算法在小波多分辨率分析中擁有極為重要的應(yīng)用，其地位和作用類似于快速傅里葉變換算法在傅里葉分析中的地位和作用。目前，小波分析仍然是國際上研究的熱點，各種新的方法和新的理論不斷的被推出。小波分析理論的這些特點使得它在時頻分析和工程應(yīng)用中得到了輝煌的發(fā)展。 本文首先介紹了小波分析的基本理論以及它的主要應(yīng)用特點，如時頻局部特性、多分辨率特性等。然后系統(tǒng)的介紹了多分辨率分析的思想，包括小波多分辨率分析和奇異值分解的多分辨率分析，并對兩種分析方法的優(yōu)劣進(jìn)行了比較。接著，因為快速分解算法在實際應(yīng)用中的重要作用，本文著重介紹了小波及小波包的快速分解算法，以及小波單子帶重構(gòu)算法等。 因為單子帶重構(gòu)算法在提取信號特征頻率成分時有很好的效果，所以本文深入的研究了單子帶重構(gòu)算法的頻域表現(xiàn)，在不斷的演算分析中，本文發(fā)現(xiàn)小波分解算法中存在著嚴(yán)重的頻率混疊現(xiàn)象，這是由于Mallat算法固有的因素造成的。即便是在單子帶重構(gòu)改進(jìn)算法中，頻率混疊現(xiàn)象仍然存在。因此，本文著重對小波分解算法產(chǎn)生頻率混疊的原因進(jìn)行了深入的剖析，并提出了一種完全抗混疊的單子帶重構(gòu)算法。此外，本文還將小波分解延伸到了小波包分析中，并且對小波包分解過程中出現(xiàn)的相似問題給出了詳細(xì)的介紹和分析。針對改進(jìn)后的單子帶重構(gòu)算法，本文把它運用到實際的故障信號中，并跟改進(jìn)前的方法進(jìn)行了對比，證實了該改進(jìn)算法的有效性。 在全文的分析推理過程中，本文除了進(jìn)行數(shù)學(xué)式子方面的推導(dǎo)外，還結(jié)合了數(shù)字信號處理方面的基礎(chǔ)知識，進(jìn)行了大量的模擬實驗。這樣，不僅可以看到抽象的理論演算，還能看到大量的直觀易懂的數(shù)據(jù)和圖像。 最后，文章對本文的工作進(jìn)行了總結(jié)，并展望了接下來的研究方向。
[Abstract]:Fourier analysis has made an outstanding contribution in signal processing, but it is defined in the whole domain, whether in the time domain or in the frequency domain. It can not analyze the signal characteristics of a certain frequency band in a certain period of time, that is, Fourier transform has no time frequency localization performance. The appearance of wavelet analysis provides an adaptive method to localize the time domain and frequency domain for signal processing. It can automatically adjust the size and shape of time-frequency window, whether it is analyzing low frequency signal or analyzing high frequency signal. In order to meet the needs of the actual analysis. At the same time, the idea of multi-resolution analysis of wavelet analysis also brings new ideas to the field of signal processing. Mallat algorithm has a very important application in wavelet multi-resolution analysis. Its position and function are similar to that of fast Fourier transform algorithm in Fourier analysis. At present, wavelet analysis is still a hot topic in the world, and many new methods and theories have been put forward. These characteristics of wavelet analysis theory make it a brilliant development in time frequency analysis and engineering applications. This paper first introduces the basic theory of wavelet analysis and its main application characteristics, such as time-frequency local characteristics, multi-resolution characteristics and so on. Then the idea of multi-resolution analysis is introduced systematically, including wavelet multi-resolution analysis and multi-resolution analysis of singular value decomposition, and the advantages and disadvantages of the two analysis methods are compared. Then, because of the important role of fast decomposition algorithm in practical application, this paper mainly introduces the fast decomposition algorithm of wavelet and wavelet packet, and the reconstruction algorithm of wavelet single subband. Because the single subband reconstruction algorithm has a good effect in extracting the characteristic frequency component of the signal, the frequency domain performance of the single subband reconstruction algorithm is deeply studied in this paper. In this paper, we find that there is a serious frequency aliasing phenomenon in wavelet decomposition algorithm, which is caused by the inherent factors of Mallat algorithm. Even in the improved single subband reconstruction algorithm, frequency aliasing still exists. Therefore, in this paper, the causes of frequency aliasing in wavelet decomposition algorithm are deeply analyzed, and a completely anti-aliasing single subband reconstruction algorithm is proposed. In addition, wavelet decomposition is extended to wavelet packet analysis, and the similar problems in the process of wavelet packet decomposition are introduced and analyzed in detail. In this paper, the improved single subband reconstruction algorithm is applied to the actual fault signal, and compared with the improved method, the effectiveness of the improved algorithm is verified. In the process of the analysis and reasoning of this paper, besides the derivation of mathematical formula, the paper also combines the basic knowledge of digital signal processing, and carries out a large number of simulation experiments. In this way, we can not only see abstract theoretical calculus, but also see a large number of intuitive and easy-to-understand data and images. Finally, the paper summarizes the work of this paper, and looks forward to the future research direction.
【學(xué)位授予單位】：重慶大學(xué)
【學(xué)位級別】：碩士
【學(xué)位授予年份】：2012
【分類號】：TH165.3;TN911.7

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