臺灣博碩士論文加值系統

本篇論文包含兩個主題:第一個主題是在討論廣義滿秩勒讓德序列(Complete Generalized Legendre Sequence)的性質及應用以及第二個主題是在討論廣義的沃爾許傅立葉轉換(Walsh-Fourier Transform)的性質及應用。
廣義滿秩勒讓德序列一開始是被定義用來解決傅立葉特徵向量的問題。我們提出的傅立葉特徵向量基於廣義滿秩勒讓德序列具有解析解、滿秩、正交、適用於任何長度的離散傅立葉轉換以及具有快速演算法。所以廣義滿秩勒讓德序列可以適合用來推導新的快速傅立葉轉換架構。奠基於實數域的研究，我們可以將研究的成果推廣到有限域當中，我們可以用有限域廣義滿秩勒讓德序列來解決數論轉換的特徵向量問題。更進一步的我們可以用廣義滿秩勒讓德序列來定義分數數論轉換以及推廣到可切換性的洗牌轉換。
沃爾許以及傅立葉轉換對於信號處理來說非常重要。我們的目的是希望用簡單的參數將上述二轉換統整起來。如此我們可以得到具有兩者優點的新轉換以及可很彈性調整的優點。我們首先將先定義離散正交轉換的特定產生方式，共軛離散正交轉換的特定產生方式以及在有限域的快速轉換推導。基於前述的成果，我們可以更進一步的去定義具有頻率順序性的廣義沃爾許傅立葉轉換以及共軛沃爾許傅立葉轉換。我們將上述定義的轉換利用在碼分多址序列設計、頻譜分析以及轉換編碼等應用。

The thesis contains two research topics: The first one is the discussion about the properties and applications of the complete generalized Legendre sequence (CGLS) and the second one is about the generalization between the Walsh-Hadamard transform (WHT) and the DFT and their properties.
The CGLS is first defined to solve the DFT eigenvector problem. The proposed CGLS based DFT eigenvectors have the advantages of closed-form solutions, completeness, orthogonality, being well defined for arbitrary N, and fast DFT expansion so that the CGLS is helpful for developing DFT fast algorithms. Based on the CGLS researches, we can extend our results to the finite field operation. That means we can also use the CGLS over finite field (CGLSF) to solve the number theoretic transform (NTT) eigenvector problem. Mean while, we can apply the CGLS and CGLSF to constructing fast DFT(NTT) algorithm, fractional number theoretic transform (FNTT) definition and the switchable perfect shuffle transform (PST) system.
The WHT and the DFT are two of the most important transforms for signal processing applications. Our purpose is to generalize these two transform by a single parameter so that the generalized transforms can not only have the advantages of the WHT and DFT but also have flexibility to some applications. We will first define the discrete orthogonal transform (DOT), conjugate symmetric discrete orthogonal transform (CS-DOT) and the fast finite field orthogonal transform (FFFOT). From the above transform, we can furthermore define the sequency ordered generalized Walsh-Fourier transform (SGWFT) and conjugate symmetric sequency ordered generalized Walsh-Fourier transform (CS-SGWFT) and show their properties and applications in CDMA sequence design, spectrum estimation and transform coding.

口試委員會審定書 #
誌謝 i
中文摘要 ii
ABSTRACT iii
CONTENTS v
LIST OF FIGURES x
LIST OF TABLES xiii
Chapter 1 Introduction 1
1.1 Complete Generalized Legendre Sequence (CGLS) 1
1.1.1 CGLS and DFT Eigenvector 1
1.1.2 CGLSF and NTT Eigenvector 1
1.1.3 Fractional Number Theoretic Transform (FNTT) 2
1.1.4 CGLS and Perfect Shuffle Transform 2
1.2 Sequency Ordered Walsh-Fourier Transform (SGWFT) 4
1.2.1 Discrete Orthogonal Transform (DOT) and SGWFT 4
1.2.2 CS-DOT and CS-SGWFT 6
1.2.3 Fast Finite Field Orthogonal Transform Without Length Constraint 7
1.3 Organization of the Dissertation 8
Chapter 2 Complete Generalized Legendre Sequence 10
2.1 Original Generalized Legendre Sequence (GLS) 10
2.1.1 Original Generalized Legendre Sequence (GLS) 10
2.1.2 Properties of GLS 11
2.2 Complete Form of Generalized Legendre Sequence 12
2.2.1 Complete Generalized Legendre Sequence (CGLS) Definition 12
2.2.2 CGLS Properties 14
2.2.3 CGLS for Arbitrary Length 16
Chapter 3 Closed Form DFT Eigenvector 19
3.1 Existing DFT Eigenvector Methods 19
3.2 DFT Eigenvectors by CGLS 21
3.2.1 Linear Combination of CGLS 21
3.2.2 Comparison with Other DFT eigenvector Methods 23
3.2.3 AM-FM Transform Relation 24
3.3 DFRFT Based on CGLS-Like DFT Eigenvectors 25
Chapter 4 Complete Generalized Legendre Sequence over Finite Field and Closed Form NTT Eigenvector 29
4.1 Number Theoretic Transform and Eigenvalue Distribution 29
4.2 Fermet Number Transform and Complex Mersenne Number Transform 30
4.3 Complete Generalized Legendre Sequence over Finite Field 32
4.4 Complete and Orthogonal NTT Eigenvectors 36
4.5 CGLSF with Arbitrary Transform Length 39
4.6 Fast Fractional NTT 41
4.6.1 Fractional NTT Based on the CGLSF Eigenvectors 41
4.6.2 Fractional Fermat Number Transform 42
4.6.3 Fractional Complex Mersenne Number Transform 43
4.6.4 Fractional New Mersenne Number Transform 44
4.6.5 Complexity Analysis 46
4.6.6 Summary on the FNTT 46
Chapter 5 DFT and NTT Implementation by CGLS and CGLSF 48
5.1 DFT Implementation by CGLS based DFT Eigenvectors 48
5.2 NTT Implementation By CGLSF based NTT Eigenvectors 53
Chapter 6 Eigenstructure of Perfect Shuffle Invariant System 58
6.1 Introduction 58
6.2 Review on Perfect Shuffle System 62
6.3 Perfect Shuffle Convolution 63
6.3.1 Perfect Shuffle Operator 63
6.3.2 Perfect Shuffle System 65
6.3.3 Perfect Shuffle Convolution Form 67
6.4 Complete Generalized Legendre Sequence Transform (CGLST) 69
6.5 Fast PST Convolution 78
6.6 Applications 79
6.6.1 Scalable Data Shuffling 79
6.6.2 Dirichlet Convolution Computation 82
6.7 Conclusions 85
Chapter 7 Discrete Orthogonal Transform (DOT) and Conjugate Symmetric Discrete Orthogonal Transform (CS-DOT) Generating Method 86
7.1 Discrete Orthogonal Transform (DOT) 88
7.1.1 Orthogonality 88
7.1.2 Radix-2 Fast Implementation Algorithm 90
7.1.3 Split-Radix Fast Implementation Algorithm 92
7.2 Conjugate Symmetric Discrete Orthogonal Transform (CS-DOT) 95
Chapter 8 Sequency Ordered Generalized Walsh-Fourier Transform 99
8.1 Review on Sequency Ordered Complex Hadamard Transform (SCHT) 100
8.2 Sequency Ordered generalized Walsh–Fourier Transform (SGWFT) 101
8.2.1 Closed-Form Expression 104
8.2.2 Properties of SGWFT 107
8.2.3 Complexity of SGWFT 111
8.3 Application of SGWFT 113
8.3.1 Signal Multiplexing 113
8.3.2 Transform Coding 116
Chapter 9 Conjugate Symmetric Sequency Ordered Generalized Walsh-Fourier Transform and Its Application 120
9.1 Review on Conjugate Symmetric Sequency Ordered Complex Hadamard Transform (CS-SCHT) 121
9.2 Conjugate Symmetric Sequency Ordered Generalized Walsh-Fourier Transform (CS-SGWFT) 123
9.3 Applications on CS-SGWFT 134
9.3.1 Spectrum Estimation 134
9.3.2 Image Interference Removal 136
Chapter 10 Fast Finite Field Orthogonal Transform (FFOT) 138
10.1 Number Theoretic Transform with Length Constraint 138
10.2 Finite Field Orthogonal Transform Generating Process 139
10.3 Fast FFOT Implementation 142
Chapter 11 Conclusion 146
REFERENCE 147
Appendix A 158
Appendix B 159
Appendix C 162
Appendix D 163
Publication List 166

[1]M. R. Schroeder, Number Theory in Science and communications, 1997, Springer-Verlag, Berlin.
[2]van Schyndel, R.G.; Tirkel, A.Z.; Svalbe, I. D., "Key independent watermark detection," Multimedia Computing and Systems, 1999. IEEE International Conference on , vol.1, no., pp.580,585 vol.1, Jul 1999
[3]J. J. Eggers, J. K. Su, “Public key watermarking by eigenectors of linear transforms” EUSIPCO 2000 Tampere, Finland, Sep 5-8 2000
[4]L. G. Hua, Introduction to Number Theory. New York: Springer-Verlag, 1982, p. 572.
[5]R. E. Blahut, Theory and Practice of Error Control Codes, Addison-Wesley, 1983.
[6]I. S. Reed and T. K. Truong, “The use of finite fields to compute convolutions,” IEEE Trans. Information Theory, vol. IT-21, no. 2, pp 208-213, Mar. 1975.
[7]D. T. Birtwistle, “The eigenstructure of the number theoretic transforms,” Signal Processing, vol. 4, pp. 287-294, Jul. 1982.
[8]Stone, H.S., “Parallel Processing with the Perfect Shuffle, ” Computers, IEEE Transactions on , vol.C-20, no.2, pp. 153- 161, Feb. 1971
[9]J. Ellis, H. Fan, and J. Shallit, “The cycles of the multiway perfect shuffle permutation,” Discrete Mathematics and Theoretical Computer Science, 5(1):169–180, 2002.
[10]M. Ma, X. Huang, B. Jiao, and Y. J. Guo, “Optimal orthogonal precoding for power leakage suppression in DFT-based systems,” IEEE Trans. Commun., vol. 59, no. 3, pp. 844-853, Mar. 2011.
[11]G. D. Mandyam, "Sinusoidal transforms in OFDM systems," IEEE Trans. Broadcast., vol. 50, no. 2, pp. 172-184, June 2004.
[12]R. Tao, X. Y. Meng, and Y. W, "Transform order division multiplexing," IEEE Trans. Signal Processing, vol. 59, no. 2, pp.598-609, Feb. 2011.
[13]A. N. Akansu and H. Agirman-Tosun, "Generalized discrete Fourier transform with nonlinear phase," IEEE Trans. Signal Processing, vol. 58, no. 9, pp.4547-4556, Sept. 2010.
[14]A. N. Akansu and H. Agirman-Tosun, "Generalized discrete Fourier transform with optimum correlations," IEEE International Conference on Acoustics Speech and Signal Processing, pp. 4054-4057, Mar. 2010.
[15]S. Bouguezel, M. O. Ahmad, and M. N. S. Swamy, "A novel transform for image compression," IEEE International Midwest Symposium on Circuits and Systems, pp. 509-512, Aug. 2010.
[16]S. Bouguezel, M. O. Ahmad, and M. N. S. Swamy, "Image encryption using the reciprocal-orthogonal parametric transform," Proceedings of IEEE International Symposium on Circuits and Systems, pp. 2542-2545, May 2010.
[17]S. Rahardja and B. J. Falkowski, “Family of unified complex Hadamard transforms,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, no. 8, pp. 1094-1100, Aug. 1999.
[18]S. Rahardja, W. Ser, and Z. Lin, “UCHT-based complex sequences for asynchronous CDMA system,” IEEE Trans. Commun., vol. 51, no. 4, pp. 618-626, Apr. 2003.
[19]A. Aung, B. P. Ng, and S. Rahardja, “Sequency-ordered complex Hadamard transform: Properties, computational complexity and applications,” IEEE Trans. Signal Processing, vol.56, no.8, pp.3562-3571, Aug. 2008.
[20]A. Aung, B. P. Ng, and S. Rahardja, “Performance of SCHT sequences in asynchronous CDMA system,” IEEE Communications Letters, vol. 11, no. 8, pp. 641-643, Aug. 2007.
[21]A. Aung, B. P. Ng, and S. Rahardja, "Conjugate symmetric sequency-ordered complex Hadamard transform," IEEE Trans. Signal Processing, vol. 57, no. 7, pp. 2582-2593, July 2009.
[22]S. Bouguezel, M. O. Ahmad, and M. N. S. Swamy, “A new class of reciprocal-orthogonal parametric transform,” IEEE Trans. Circuits Syst. I, Regul. Pap., vol. 56, no. 4, pp. 795-805, Apr. 2009.
[23]S. Bouguezel, M. O. Ahmad, and M. N. S. Swamy, "New parametric discrete Fourier and Hartley transforms, and algorithms for fast computation," IEEE Trans. Circuits Syst. I, Regul. Pap., vol. 58, no. 3, pp. 562-575, Mar. 2011.
[24]Z. Chen, M. H. Lee, and G. Zeng, "Fast cocyclic Jacket transform," IEEE Trans. Signal Processing, vol. 56, no. 5, pp. 2143-2148, May 2008.
[25]M. H. Lee and K. Finlayson, “A simple element inverse Jacket transform coding,” IEEE Signal Process. Lett., vol. 14, no. 5, pp. 325-328, May 2007.
[26]Jiasong Wu; Lu Wang; Guanyu Yang; Senhadji, L.; Limin Luo; Huazhong Shu; , "Sliding Conjugate Symmetric Sequency-Ordered Complex Hadamard Transform: Fast Algorithm and Applications," Circuits and Systems I: Regular Papers, IEEE Transactions on , vol.59, no.6, pp.1321-1334, June 2012.
[27]J. H. McClellan, “Eigenvalue and eigenvector decomposition of the discrete Fourier transform,” IEEE Trans. Audio Electroacoust., vol. 20, pp. 66-74, 1972.
[28]B. W. Dickinson and K. Steiglitz, “Eigenvectors and functions of the discrete Fourier transform,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 30, pp. 25-31, Feb. 1982.
[29]C. Candan, M. A. Kutay, and H. M. Ozaktas, “The discrete fractional Fourier transform,” IEEE Trans. Signal Processing, vol. 48, pp. 1329-1337 May 2000.
[30]N. M. Atakishiyev, L. E. Vicent, and K. B. Wolf, “Continuous vs. discrete fractional Fourier transforms,” v.107, n.1, pp.73-95, 1999.
[31]M. L. Mehta, “Eigenvalues and eigenvectors of the finite Fourier transform,” Journal. Math. Phys. 28, no. 4, pp.781-785 April. 1987
[32]F. A. Grunbaum, “The eigenvectors of the discrete Fourier transform: a version of the Hermite functions,” Journal of Mathematical analysis and applications, vol. 88, p.355-363, 1982
[33]S. Clary and D. H. Mugler, “Shifted Fourier matrices and their tridiagonal commuters”, SIAM J. Matrix Anal. Appl., vol. 24, no. 3, pp. 809-821, March 2003.
[34]D. H. Mugler and S. Clary, “Discrete Hermite functions and the fractional Fourier transform”, Proc. International Conferences on Sampling Theory and Applications, pp. 303-308, 2001.
[35]Soo-Chang Pei; Wen-Liang Hsue; Jian-Jiun Ding, "Discrete Fractional Fourier Transform Based on New Nearly Tridiagonal Commuting Matrices," Signal Processing, IEEE Transactions on , vol.54, no.10, pp.3815,3828, Oct. 2006
[36]Soo-Chang Pei; Yun-Chiu Lai, "Signal Scaling by Centered Discrete Dilated Hermite Functions," Signal Processing, IEEE Transactions on , vol.60, no.1, pp.498,503, Jan. 2012
[37]Soo-Chang Pei; Wen-Liang Hsue; Jian-Jiun Ding, "DFT-Commuting Matrix With Arbitrary or Infinite Order Second Derivative Approximation," Signal Processing, IEEE Transactions on , vol.57, no.1, pp.390,394, Jan. 2009
[38]T. Erseghe, and G..Cariolaro, “An orthonormal class of exact and simple DFT eigenvectors with a high degree of symmetry,” IEEE Trans. Signal Processing, vol. 51, pp. 2527-2539 Oct. 2003.
[39]M. S. Pattichis, A. C. Bovik, J. W. Havlicek, and N. D. Sidiropoulos, “Multidimensional orthogonal FM transforms”, IEEE Trans. Image Processing, vol. 10, no. 3, pp. 448-464, March 2001.
[40]S. C. Pei and M. H. Yeh, “Discrete fractional Fourier transform,” in Proc. IEEE Int. Symp. Circuits Syst., May 1996, pp. 536-539.
[41]S. C. Pei, M. H. Yeh, and C. C Tseng, “Discrete fractional Fourier transform based on orthogonal projections,” IEEE Trans. Signal Processing, vol. 47, pp. 1335-1348, 1999.
[42]J. G. Vargas-Rubio and B. Santhanam, “On the multiangle centered discrete fractional Fourier transform”, IEEE Signal Processing Letters, vol. 12, no. 4, pp. 273-276, Apr. 2005.
[43]J. G. Vargas-Rubio and B. Santhanam, “An improved spectrogram using the multiangle centered discrete fractional Fourier transform”, vol. 4, pp. 505-508, ICASSP, 2005.
[44]S. C. Pei, C. C. Wen, and J. J. Ding, “Closed-form orthogonal DFT eigenvectors generated by complete generalized Legendre sequence,” IEEE Trans. Circuits Syst. I, vol. 55, no. 17, pp. 3469-3479, Dec. 2008.
[45]C. Candan, "On higher order approximations for Hermite - Gaussian functions and discrete fractional Fourier transforms", IEEE Signal Process. Lett., vol. 14, no. 10, pp. 699-702, 2007.
[46]S. Boussakta and A. G. J. Holt, “New transform using the Mersenne numbers,” IEE Proc. Vis. Image Signal Process., vol. 142, no. 6, pp. 381-388, Dec 1995.
[47]Y. Jakop, A, S, Madhukumar, and A. B. Premkumar, “A robust symmetrical number system based parallel communication system with inherent error detection and correction,” IEEE Trans. Wireless Communication, vol. 8, no. 6, pp. 2742-2747, June 2009.
[48]D. A. Hejhal, Emerging Applications of Number Theory, Springer, New York, 1999.
[49]Stone, H.S., “Parallel Processing with the Perfect Shuffle, ” Computers, IEEE Transactions on , vol.C-20, no.2, pp. 153- 161, Feb. 1971
[50]J. Ellis, H. Fan, and J. Shallit, “The cycles of the multiway perfect shuffle permutation,” Discrete Mathematics and Theoretical Computer Science, 5(1):169–180, 2002.
[51]Hluchyj, M.G.; Karol, M.J.; , "ShuffleNet: an application of generalized perfect shuffles to multihop lightwave networks," INFOCOM ''88. Networks: Evolution or Revolution, Proceedings. Seventh Annual Joint Conference of the IEEE Computer and Communcations Societies, IEEE , vol., no., pp.379-390, 27-31 Mar 1988
[52]Ben-Asher, Y.; Egozi, D.; Schuster, A.; , "2-D SIMD Algorithms In The Perfect Shuffle Networks," Computer Architecture, 1989. The 16th Annual International Symposium on , vol., no., pp.88-95, 28 May-1 Jun 1989
[53]Godwin, D.P.; Carey, C.D.; Song, S.H.; Selviah, D.R.; Midwinter, J.E.; , "Perfect shuffle interconnections using Fresnel computer generated holograms in a planar-optic configuration," Holographic Systems, Components and Applications, 1993., Fourth International Conference on , vol., no., pp.15-20, 13-15 Sep 1993
[54]Corinthios, M.J.; , "Optimal parallel and pipelined processing through a new class of matrices with application to generalized spectral analysis," Computers, IEEE Transactions on , vol.43, no.4, pp.443-459, Apr 1994
[55]Jarvinen, T.S.; Takala, J.H.; Akopian, D.A.; Saarinen, J.P.P.; , "Register-based multi-port perfect shuffle networks," Circuits and Systems, 2001. ISCAS 2001. The 2001 IEEE International Symposium on , vol.4, no., pp.306-309 vol. 4, 6-9 May 2001
[56]J. C. Lin; M. Hsieh; M. J. F. Chiang; S. Y. Mao; C. Yu; S. J. Chen; Y. H. Hu; , "Perfect shuffling for cycle efficient puncturer and interleaver for software defined radio," Circuits and Systems (ISCAS), Proceedings of 2010 IEEE International Symposium on , vol., no., pp.3965-3968, May 30 2010-June 2 2010
[57]Schiano, L.; Lombardi, F.; , "On the test and diagnosis of the perfect shuffle," Defect and Fault Tolerance in VLSI Systems, 2003. Proceedings. 18th IEEE International Symposium on , vol., no., pp. 97- 104, 3-5 Nov. 2003
[58]Davio, M.; , "Kronecker products and shuffle algebra," Computers, IEEE Transactions on , vol.C-30, no.2, pp.116-125, Feb. 1981
[59]Suleiman, A.; Hussein, A.; Bataineh, K.; Akopian, D.; , "Scalable FFT architecture vs. multiple pipeline FFT architectures — Hardware implementation and cost," Systems, Man and Cybernetics, 2009. SMC 2009. IEEE International Conference on , vol., no., pp.3792-3796, 11-14 Oct. 2009
[60]Takala, J.; Akopian, D.; Astola, J.; Saarinen, J.; , "Scalable interconnection networks for partial column array processor architectures," Circuits and Systems, 2000. Proceedings. ISCAS 2000 Geneva. The 2000 IEEE International Symposium on , vol.4, no., pp.513-516 vol.4, 2000
[61]Hussein, A.; Suleiman, A.; Kerkiz, N.; Akopian, D.; , "Implementation of scalable interconnect networks for data reordering used in Discrete Trigonometric Transforms (DTT)," IC Design and Technology, 2009. ICICDT ''09. IEEE International Conference on , vol., no., pp.139-142, 18-20 May 2009
[62]Ray, G.A.; , "VLSI architectures for Dirichlet arithmetic," Acoustics, Speech, and Signal Processing, 1992. ICASSP-92., 1992 IEEE International Conference on , vol.4, no., pp.589-592 vol.4, 23-26 Mar 1992
[63]M. Ma, X. Huang, B. Jiao, and Y. J. Guo, “Optimal orthogonal precoding for power leakage suppression in DFT-based systems,” IEEE Trans. Commun., vol. 59, no. 3, pp. 844-853, Mar. 2011.
[64]G. D. Mandyam, "Sinusoidal transforms in OFDM systems," IEEE Trans. Broadcast., vol. 50, no. 2, pp. 172-184, June 2004.
[65]R. Tao, X. Y. Meng, and Y. W, "Transform order division multiplexing," IEEE Trans. Signal Processing, vol. 59, no. 2, pp.598-609, Feb. 2011.
[66]A. N. Akansu and H. Agirman-Tosun, "Generalized discrete Fourier transform with nonlinear phase," IEEE Trans. Signal Processing, vol. 58, no. 9, pp.4547-4556, Sept. 2010.
[67]A. N. Akansu and H. Agirman-Tosun, "Generalized discrete Fourier transform with optimum correlations," IEEE International Conference on Acoustics Speech and Signal Processing, pp. 4054-4057, Mar. 2010.
[68]S. Bouguezel, M. O. Ahmad, and M. N. S. Swamy, "A novel transform for image compression," IEEE International Midwest Symposium on Circuits and Systems, pp. 509-512, Aug. 2010.
[69]S. Bouguezel, M. O. Ahmad, and M. N. S. Swamy, "Image encryption using the reciprocal-orthogonal parametric transform," Proceedings of IEEE International Symposium on Circuits and Systems, pp. 2542-2545, May 2010.
[70]S. Rahardja and B. J. Falkowski, “Family of unified complex Hadamard transforms,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, no. 8, pp. 1094-1100, Aug. 1999.
[71]S. Rahardja, W. Ser, and Z. Lin, “UCHT-based complex sequences for asynchronous CDMA system,” IEEE Trans. Commun., vol. 51, no. 4, pp. 618-626, Apr. 2003.
[72]A. Aung, B. P. Ng, and S. Rahardja, “Sequency-ordered complex Hadamard transform: Properties, computational complexity and applications,” IEEE Trans. Signal Processing, vol.56, no.8, pp.3562-3571, Aug. 2008.
[73]A. Aung, B. P. Ng, and S. Rahardja, “Performance of SCHT sequences in asynchronous CDMA system,” IEEE Communications Letters, vol. 11, no. 8, pp. 641-643, Aug. 2007.
[74]A. Aung, B. P. Ng, and S. Rahardja, "Conjugate symmetric sequency-ordered complex Hadamard transform," IEEE Trans. Signal Processing, vol. 57, no. 7, pp. 2582-2593, July 2009.
[75]F. Schipp, W. R. Wade, P. Simon, and J. Pal., Walsh Series. An Introduction to Dyadic Harmonic Analysis, Adam Hilger Ltd., Bristol, 1990.
[76]F. Schipp and W. R. Wade, Norm convergence and summability of Fourier series with respect to certain product systems. in: Approximation Theory. New York: Pure Appl Math Dekker, vol. 138, 1992, pp. 437–452.
[77]G. Gat, On (C,1) summability for Vilenkin-like systems, Stud. Math., vol. 144, 2001, pp. 101-120.
[78]G. Gat, On (C, 1) summability of integrable functions with respect to theWalsh-Kaczmarz system, Stud. Math., vol. 130 1998, pp. 135-148.
[79]G. Gat and K. Nagy, Cesaro summability of the character system of the p-series field in the Kaczmarz rearrangement, Anal. Math., vol. 28, 2002, pp. 1-36.
[80]U. Goginava, The maximal operator of the Fejer means of the character system of the p-series field in the Kaczmarz rearrangement, Publ. Math., vol. 71, 2007, pp. 43-55.
[81]Z. Chuanzhou and Z. Xueying, On the Fejer kernel functions of two-dimensional Walsh systems by Kaczmarz rearrangement, Proceedings of the 2nd International Conference on Modelling and Simulation, vol. 5, 2009, pp. 421-424.
[82]I. V. Polyakov, (C; 1)-Summation of Fourier series over rearranged Vilenkin system, Moscow University Mathematics Bulletin vol. 65, 2010, pp. 140-147.
[83]F. Schipp and W. R. Wade, Fast Fourier transforms on binary fields, Approximation Theory and Its Applications, vol. 14, 1998, pp. 91-100.
[84]S.C. Pei, C.C. Wen, and J.J. Ding, “Sequency-ordered generalized Walsh–Fourier transform”, Signal Processing, vol. 93, Issue 4, pp. 828-841, Apr. 2013
[85]S. Bouguezel, M. O. Ahmad, and M. N. S. Swamy, “A new class of reciprocal-orthogonal parametric transform,” IEEE Trans. Circuits Syst. I, Regul. Pap., vol. 56, no. 4, pp. 795-805, Apr. 2009.
[86]S. Bouguezel, M. O. Ahmad, and M. N. S. Swamy, "New parametric discrete Fourier and Hartley transforms, and algorithms for fast computation," IEEE Trans. Circuits Syst. I, Regul. Pap., vol. 58, no. 3, pp. 562-575, Mar. 2011.
[87]Z. Chen, M. H. Lee, and G. Zeng, "Fast cocyclic Jacket transform," IEEE Trans. Signal Processing, vol. 56, no. 5, pp. 2143-2148, May 2008.
[88]M. H. Lee and K. Finlayson, “A simple element inverse Jacket transform coding,” IEEE Signal Process. Lett., vol. 14, no. 5, pp. 325-328, May 2007.
[89]S. H. Tsai, Y. P. Lin, and C. C. J. Kuo, “MAI-free MC-CDMA systems based on Hadamard-Walsh codes,” IEEE Trans. Signal Processing, vol. 54, no. 8, pp. 3166-3179, Aug. 2006.
[90]C. S. Burrus and T. W. Parks, DFT/FFT and Convolution Algorithms: Theory and Implementation, John Wiley &; Sons, New York, 1985.
[91]D. V. Sarwate and M. B. Pursley, “Cross correlation properties of pseudorandom and related sequences,” Proc. IEEE, vol. 68, no. 5, pp. 593-619, May 1980.
[92]N. Tayem, H. M. Kwon, M. Seunghyun, and H. K. Dong, "Covariance matrix differencing for coherent source DOA estimation under unknown noise field," Vehicular Technology Conference, pp. 1-5, Sept. 2006.
[93]K. Wang, Y. Zhang, and D. Shi, “Novel algorithm on DOA estimation for correlated sources under complex symmetric Toeplitz noise,” Journal of Systems Engineering and Electronics, vol. 19, no. 5, pp. 902-906, Aug. 2008.
[94]D. R. Fuhrmann and T. A. Barton, "New results in the existence of complex covariance estimates," Conference Record of the 26th Asilomar Conference on Signals, Systems and Computers, vol. 1, pp. 187-191, Oct. 1992.
[95]D. R. Fuhrmann and M. I. Miller, "On the existence of positive-definite maximum-likelihood estimates of structured covariance matrices," IEEE Trans. Inf. Theory, vol. 34, no. 4, pp. 722-729, Jul. 1988.
[96]M. J. Turmon and M. I. Miller, "Maximum-likelihood estimation of complex sinusoids and Toeplitz covariances," IEEE Trans. Signal Processing, vol. 42, no. 5, pp. 1074-1086, May 1994.