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研究生： 陳奇宏
Chen, Che-Hong
論文名稱： 高效率弦波轉換之遞迴演算法
Effective Recursive Algorithms for Discrete Sinusoidal Transforms
指導教授： 楊家輝
Yang, Jar-Ferr
劉濱達
Liu, Bin-Da
學位類別： 博士
Doctor
系所名稱： 電機資訊學院 - 電機工程學系
Department of Electrical Engineering
論文出版年： 2005
畢業學年度： 93
語文別： 英文
論文頁數： 125
中文關鍵詞： 遞迴 、弦波轉換
外文關鍵詞： Recursive, Discrete Sinusoidal Transforms
相關次數： 點閱：84  下載：2 
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    • 本論文主要是針對一般的離散弦波轉換提出高效率之遞迴架構演算法。由於無限脈衝響應(IIR)濾波器架構花費較多的計算時間完成遞迴運算，同時造成嚴重的取捨誤差，對於多維度的計算，更需搭配暫存記憶體來完成計算，因此，針對以上缺點與IIR濾波器架構的特性，我們做深入的分析與討論，進一步對一維與多維度離散弦波轉換提出高效率之遞迴架構演算法。
    一維離散弦波轉換方面，首先我們提出定係數遞迴架構演算法，利用自訂的modulo技巧，此演算法可用來計算任意長度離散弦波轉換。基於原始資料的交換排列、符號改變與零值內插，此架構比其他定係數演算法所需的計算迴圈更少。在有限字元長度的計算器，適當調整遞迴濾波器的係數，可得到更準確的結果。此外，我們又提出高輸出率遞迴架構演算法，利用對摺的技巧，可減少遞迴運算的次數。實驗結果顯示，因為所提出的定係數與高輸出率遞迴架構演算法需要較少的計算迴圈，同時擁有較高的輸出率，所以可得到較準確的結果。此外，所提架構可適用於一維DCT/IDCT、MDCT/IMDCT、分頻合成濾波轉換(Subband synthesis filter)。
    多維離散弦波轉換方面，我們提出高效率之精簡遞迴架構，利用三角函數公式與數論，由探討形成一完全剩餘系統的週期性轉換基底，發展出需要較少之遞迴運算迴圈的精簡離散餘弦轉換遞迴架構演算法。此演算法最大的特點是不需要暫存記憶體即可直接計算多維度的離散餘弦轉換，因此可以在較少的功率消耗與存取時間的優點下，得到精確的結果。模擬結果亦顯示所提的精簡遞迴架構演算法比用row-column方法實現的一維遞迴演算法有較高的PSNR值。此外，規則的架構有利於VLSI的實現，利用相似的推導原理，此演算法更可以發展成多維度離散正弦轉換遞迴架構。

    In this dissertation, we propose the effective recursive algorithms for discrete sinusoidal transforms, which can be realized as the infinite impulse response (IIR) filters. Regarding to the IIR filtering structure, the longer computational time, larger round-off error and the requirement of transposition memory for multi-dimensional discrete sinusoidal transform are the drawbacks. According to the properties of the IIR filter, some techniques are developed to overcome above disadvantages. Furthermore, the effective recursive algorithms for one-dimensional and multi-dimensional discrete sinusoidal transforms are respectively addressed.
    For one-dimensional discrete sinusoidal transforms, such as MDCT/IMDCT, DCT/IDCT, DST/IDST and subband analysis/synthesis filter, fixed-coefficient recursive structure and high-throughput recursive structure are proposed to provide the high effective and more accurate IIR filtering structures. Applying the proposed modified modulo technique, the fixed-coefficient recursive structures for computing the general length discrete sinusoidal transforms, which are simplified and converted into the simpler DCT, are developed. Based on permutation, sign change and zero insertion of original inputs, the proposing filtering structures required fewer recursive cycles than the previous fixed-coefficient algorithms. Furthermore, by selecting the optimal coefficient, the IIR filtering structures can achieve more accurate results. On the other hand, applying the data-folding technique, the high-throughput recursive formulas, which require fewer recursive cycles to complete computation, are addressed. The proposed fixed-coefficient and high-throughput recursive algorithms require fewer computational cycles and achieve higher data throughput per transformation than the existing methods. The simulation results show that fewer computational cycles and higher data throughput will effectively result in smaller round-off error.
    For multi-dimensional discrete sinusoidal transform, such as multi-dimensional DCT/IDCT, condensed recursive structures are proposed to provide effective and accurate IIR filtering structures. Using trigonometric identities and number theory, the condensed recursive structures requiring fewer recursive loops are developed from exploration of periodicity embedded in transform bases. The most significant feature of the proposed algorithm is that the transformation is computed directly without using any transposition memory. Hence, the proposed algorithm achieves more accurate results with less power consumption and smaller access time. Simulation results verify that the proposed algorithm provides higher average PSNR’s and throughput than other algorithms using the traditional row-column method. In addition, due to the less computational complexity, the proposed algorithm is suitable for VLSI implementation. Using the similar derivation, the proposed algorithm can be applied to obtain the multi-dimensional DST/IDST recursive structures.

    Contents List of Figures iii List of Tables v Abstract vi CHAPTER 1 Introduction 1 1.1 Motivation 1 1.2 Background 3 1.2.1 Fast Algorithm Design 4 1.2.2 Recursive Architecture Design 5 1.3 Organization of the Dissertation 6 CHAPTER 2 Recursive Structures for Discrete Sinusoidal Transforms 9 2.1 Introduction 9 2.2 Goertzel’s Recursive Structure DFT Algorithm 10 2.3 Generic Recursive Structure 11 2.4 Accuracy Consideration 15 CHAPTER 3 Recursive Algorithms for One-Dimensional Discrete Sinusoidal Transforms 17 3.1 Introduction 17 3.2 Recursive Structures with Fixed Coefficients 19 3.2.1 Fixed-Coefficient Recursive Structure for Discrete Sinusoidal Transform 19 3.2.2 Selected Filter Coefficient 24 3.3 Recursive Structures with Higher Output Throughput and Fewer Recursive Cycles 29 3.3.1 High-Throughput Recursive Structures for DCT 29 3.3.2 High-Throughput Recursive Structures for IDCT 36 3.3.3 High-Throughput Recursive Structure for MDCT 47 3.3.4 High-Throughput Recursive Structure for IMDCT 50 3.4 Complexity Comparisons and Discussion 57 3.5 Summary 59 CHAPTER 4 Recursive Algorithms for Multi-dimensional Discrete Sinusoidal Transforms 65 4.1 Introduction 65 4.2 Condensed Recursive Structures for M-D DCT 66 4.2.1 Condensed-kernel Bases Conversion of M-D DCT to 1-D DCT 66 4.2.2 IIR filtering Structures 72 4.3 Condensed Recursive Structures for M-D IDCT 75 4.4 Design Examples 77 4.4.1 4×4 2-D DCT 77 4.4.2 Hardware Design for 4×4 2-D DCT 80 4.4.3 8×8×N 3-D DCT 83 4.4.4 Hardware Design for 8×8×N 3-D DCT 88 4.5 Complexity Comparisons and Discussion 92 CHAPTER 5 Conclusions and Future Works 103 5.1 Conclusions 103 5.2 Further work 105 References 107 Publication List 117 List of Figures Fig. 2. 1 The recursive structure for Goertzel’s algorithm. 11 Fig. 2. 2 The generic recursive structure for discrete cosine transform. 13 Fig. 2. 3 The generic recursive structure for discrete sine transform. 15 Fig. 2. 4 The allocation of poles in Goertzel’s algorithm of DFT. 16 Fig. 3. 1 The fixed-coefficient recursive kernel as ab is odd. 22 Fig. 3. 2 The fixed-coefficient recursive kernel as ab is even. 23 Fig. 3. 3 The allocation of poles of the proposed algorithm. 24 Fig. 3. 4 The modified fixed-coefficient recursive kernel with scalable q factor as ab is odd. 26 Fig. 3. 5 The modified fixed-coefficient recursive kernel with scalable q factor as ab is even. 26 Fig. 3. 6 The allocation of poles of the proposed algorithm with q-factor. 27 Fig. 3. 7 The accuracy performance with different q factor for the DCT-III (a = 1, b = 0). 28 Fig. 3. 8 The proposed IFR kernels for computation of DCT coefficients (a) kernel for even coefficient (b) kernel for odd coefficient. 32 Fig. 3. 9 The proposed IFR structure for computation of two DCT coefficients. 32 Fig. 3. 10 The proposed DIFR kernels for computing DCT architecture (a) kernel for even coefficient (b) kernel I for odd coefficient (c) kernel II for odd coefficient 35 Fig. 3. 11 The proposed DIFR structure for computation of two DCT coefficients. 35 Fig. 3. 12 The proposed OFR kernels for computing IDCT architecture (a) kernel for even-index sequence (b) kernel for odd-index sequence. 39 Fig. 3. 13 The proposed OFR structure for computation of two IDCT coefficients. 40 Fig. 3. 14 The proposed DOFR kernels for computing IDCT architecture (a) kernel for even-index sequence (b) kernel I for odd-index sequence (c) kernel II for odd-index sequence. 46 Fig. 3. 15 The proposed DOFR structure for computation of four IDCT coefficients. 46 Fig. 3. 16 Recursive structure of the MDCT. 50 Fig. 3. 17 Recursive structure of the IMDCT. 56 Fig. 3. 18 Accuracy comparison for the DCT-III with N = 8. 61 Fig. 4. 1 Recursive kernel for condensed 1-D DCT. 74 Fig. 4. 2 Recursive kernel for condensed 1-D DST. 74 Fig. 4. 3 Recursive kernel for condensed 1-D DCT/DST. 74 Fig. 4. 4 Realization of the proposed 4×4 2-D DCT condensed recursive structure. 81 Fig. 4. 5 Implementation of Coefficient Decomposition Unit. 82 Fig. 4. 6 The structure of the Condensed Index Generator. 82 Fig. 4. 7 Hardware architecture of 8×8×N 3-D DCT. 89 Fig. 4. 8 Implementation of Coefficient Decomposition Unit. 90 Fig. 4. 9 Implementation of Index Counter. 91 Fig. 4. 10 Structure of Condensed Index Generator. 92 Fig. 4. 11 Accuracy performance in SNR for 16×16 2-D DCT. 94 List of Tables Table 3. 1 Relationships between the input parameters and the desired transforms. 23 Table 3. 2 Number of recursive cycles for N inputs recursive structures. 60 Table 3. 3 Accuracy performance in PSNR’s (dB) of DCT-III fixed-coefficient recursive structures in finite-word length machines. 60 Table 3. 4 The symbols with respect to various methods. 60 Table 3. 5 Hardware overhead and processing speed comparisons among different architectures for N-point DCT implementation. 62 Table 3. 6 Hardware overhead and processing speed comparisons among different architectures for N-point IDCT implementation. 62 Table 3. 7 Hardware comparisons for MDCT implementation. 63 Table 3. 8 Algorithms comparisons for various IMDCT implementation. 63 Table 4. 1 The index mapping table for = 2n1 + n2 mod 4. 97 Table 4. 2 The index mapping table for = 2n1 - n2 mod 4. 97 Table 4. 3 Index mapping table for = 6n1 + 15n2 +8n3 mod 24 while n1 = 0. 97 Table 4. 4 Numbers of multiplication and addition operations for 4×4 DCT. 98 Table 4. 5 Numbers of recursive cycles for N×N DCT recursive structures. 98 Table 4. 6 Numbers of recursive cycles for N×N IDCT recursive structures. 99 Table 4. 7 Comparison of 8×8×8 DCT/IDCT recursive structures. 100 Table 4. 8 Accuracy performance of finite word-length machine in PSNR (dB) for various 8×8×8 DCT/IDCT realization methods. 101

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