臺灣博碩士論文加值系統

S 轉換是一個功能強大且具有漸進式解析度的時頻域分佈演算法。它已經被成功地廣泛應用於許多不同的領域。由於它是線性的，所以只要乘上一個遮罩，就可以輕易地利用它在時頻域上進行濾波了。目前存在著許多不同的反Ｓ轉換，各自有不同的性質。其中最廣泛使用的是由Stockwell等人所提出來的方法。它具有很高的效率，但是時間解析度卻差強人意。而由Schimmel和Gallart所提出的方法雖然改進了這個缺點，但卻犧牲了頻率解析度，並且會造成還原上的誤差。在本論文中，我們提出了新的反Ｓ轉換，其中加入了等效濾波器，以消除上述兩個方法在濾波時所造成的失真。此外，我們也推導出Ｓ轉換的轉換矩陣，並據以推導出兩個具有最小平方誤差的反Ｓ轉換。其中第一個方法是對整個時頻域頻譜去計算最小平方誤差，而第二個方法只針對局部時頻域做考量，提供更有彈性的應用。實驗結果顯示，比起前人的方法，我們所提出的反Ｓ轉換具有更好、更穩定的時頻域濾波器效能。
近來研究人員發現數位Ｓ轉換含有在時域和頻域不一致的問題，這可能會導致不可靠的時頻域分佈資訊以及造成研究上的困難。針對這個問題，到目前為止所提出的解決辦法還有著一些缺點而無法讓人滿意。在本論文中，我們提出一個新的數位Ｓ轉換，它採用折疊式的高斯函數，可以有效解決這個問題。這個新個數位Ｓ轉換在時域和頻域上能夠達到一致，並提供更可靠的時頻域分佈資訊。此外，它也完整的繼承了連續Ｓ轉換的許多重要特性。
本論文中的另一重要貢獻是使得Ｓ轉換中的能量更加集中，以提高解析度。Djurović等人根據能量集中度測量法提出了最佳化高斯函數寬度的方法。然而，這個方法只對中高頻的信號有效，而無法對低頻部分產生助益。在本論文中，我們提出一個新的方法，有效改善這個缺點。經過實驗比較後，可以發現經由我們的方法所得到的能量集中度，更勝於Djurović等人的方法以及傳統的Ｓ轉換。

The S transform is a powerful linear time-frequency distribution with a progressive resolution. It has been shown useful in various fields of applications. Since it is linear, it filters efficiently in a time-frequency domain by multiplying a mask function. There exist several different inverse algorithms, which result in different filtering effects. The conventional inverse S transform proposed by Stockwell et al. is efficient but suffers from time leakage during filtering. The recent algorithm proposed by Schimmel and Gallart has better time localization during filtering but suffers from a reconstruction error and the frequency leakage. In this dissertation, we propose two new inverse S transforms with equalization filters that compensate the distortion resulted from the previous two methods during filtering. Besides, we also derive the transformation matrices of the S transform and two novel least square inverse algorithms. The first one minimizes the global mean squared error of the entire time-frequency spectrum, and the second one considers only the specific interesting time-frequency regions and is more flexible. Experimental results show that the proposed inverse S transforms provide more stable and better performance in time-frequency filtering than the existing ones.
Recently researchers noticed that the conventional discrete S transforms are not equivalent in the time and the frequency domains, which may result in unreliable time-frequency information and confuse researchers. The solution thus far has some drawbacks and is unsatisfactory. In this dissertation, a new discrete S transform adopting the folded windows is proposed to eliminate the side effects of discretizing. This new discrete S transform has the theoretical importance that the time version and frequency version are equivalent and therefore provides more reliable information than previous solutions. Furthermore, it inherits the properties from the continuous S transform more completely.
Another important improvement we made to the S transform is to enhance the energy concentration. Based on the concentration measure, Djurović et al. proposed a method to optimize the window width in the S transform. However, it is found that although their method performs well for the high and middle frequency signals, it may fail for the low frequency ones. In this dissertation, a new method, which is more flexible than the previous one, is proposed to deal with this problem. Comparison of these two methods for energy concentration enhancement is also provided. Experimental result shows that the proposed method achieves higher energy concentration in comparison to the previous one and the original ST.

Chinese Abstract iii
Abstract v
List of Figures xi
List of Tables xiii
Chapter 1. Introduction 1
1.1 Motivation 1
1.2 Review of S Transform 5
1.2.1. Generalized S Transform 5
1.2.2. Stockwell et al.’s IST 7
1.2.3. Schimmel and Gallart’s IST 7
1.3 Dissertation Organization 8
Chapter 2. Inverse S Transform with Equalization Filter 11
2.1 Modified IST 11
2.2 Time-EQ IST 15
2.3 Freq-EQ IST 25
2.4 Experimental results 29
2.4.1. Correcting the distortion of Time IST 29
2.4.2. Time localization in filtering 29
2.4.3. Frequency localization in filtering 31
2.4.4. Noise reduction 33
2.4.5. Adaptive time-frequency filtering 34
Chapter 3. Inverse S Transform with Least Square Error 37
3.1 Matrix formulation of ST 37
3.2 Global Least Square IST 39
3.3 Local Least Square IST 40
3.4 Experimental Results 46
3.4.1. Removing two chirp noises 46
3.4.2. Removing the white noise 50
3.4.3. Removing the 1/f noise 53
3.4.4. Removing three chirp noises 55
Chapter 4. Elimination of Side Effect Using Folded Windows 57
4.1 Side Effects in Discrete S Transform 57
4.2 S Transform with Folded Window 61
4.3 Experimental Results 66
Chapter 5. Energy Concentration Enhancement Using Window Width Optimization 73
5.1 Djurović et al.’s optimization method 74
5.2 Our optimization method 76
5.3 Experimental Results 78
Chapter 6. Conclusion 83
6.1 Concluding Remarks 83
6.2 Future works 85
Bibliography 87

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