Filter Banks

Filter banks allow signals to be decomposed into subbands. In this way, parallel powerful processing can be easily applied. Also, the decomposition paves the way for signal compression procedures. Due to these reasons, the interest on filter banks has significantly grown along years, so today there is large body of theory on this matter. This chapter is also important for other reasons, since it serves as one of the pertinent ways for introducing wavelets, as it will be confirmed in the next chapter and other parts of this book. The main topic in relation with filter banks and wavelets is ‘perfect reconstruction’, which is treated in detail. Some interesting aspects in this chapter are lattice structures, allpass filters, the discrete cosine transform (DCT), JPEG, and watermarking.

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References

  1. T. Aach, Fourier, block and lapped transforms, in Advances in Imaging and Electron Physics, ed.by P.W. Hawkes (Academic Press, 2003) Google Scholar
  2. N. Ahmed, T. Natarajan, K.R. Rao, Discrete cosine transform. IEEE T. Comput. 90–93, 1974 Google Scholar
  3. I. Balasingham, T. Ramstad, J. Lervik, Survey of odd and even length filters in tree-structured filter banks for subband image compression, in Proceedings of the IEEE International Conference Acoustics, Speech and Signal Processing, vol. l4, pp. 3073–3076 (1997) Google Scholar
  4. C. Brislawn, A simple lattice architecture for even-order linear-phase perfect reconstruction filter banks, in Proceedings of the IEEE International Symposium Time-Frequency and Time-Scale Analysis, pp 124–127 (1994) Google Scholar
  5. W.H. Chen, C.H. Smith, S.C. Fralick, A fast computational algorithm for the discrete cosine transform. IEEE T. Commun. l25(9):1004–1009 (1977) Google Scholar
  6. J. Cox, M.L. Miller, J.A. Boom, J. Fridrich, T. Kalker, Digital Watermarking and Stegonography (Morgan Kaufmann, 2007) Google Scholar
  7. H.V. Dwivedi, Design of JPEG Compressor (National Institute of Technology, Rourkela, India, 2009) Bachl. thesis Google Scholar
  8. A. Fettweis, H. Levin, A. Sedlmeyer, Wave digital lattice filters. Intl. J. Circuit Theory Appl. 2, 203–211 (1974) Google Scholar
  9. S. Foucart, Linear Algebra and Matrix Analysis (Math 504 Lectures Notes, Drexel University, 2010). http://www.math.drexel.edu/foucart/teaching.htm
  10. F. Galijasevic, Allpass-Based Near-Perfect-Reconstruction Filter Banks. PhD thesis (Cristian-Albrechts University, Kiel, Germany, 2002) Google Scholar
  11. X. Gao, T.Q. Nguyen, G. Strang, On factorization of M-channel paraunitary filterbanks. IEEE T. Sign. Process. 49(7), 1433–1446 (2001) Google Scholar
  12. H. Haberdar, Discrete Cosine Transform Tutorial (2013). www.haberdar.org/Discrete-Cosine_Transform-Tutorial.htm
  13. C. Herley, M. Vetterli, Wavelets and recursive filter banks. IEEE T. Sign. Process. 41(8), 2536–2556 (1993) Google Scholar
  14. S.A. Khayam, The Discrete Cosine Transform (DCT): Theory and Applications. (Michigan State University, 2003). Notes of the ECE 802-602 course Google Scholar
  15. J. Kliewer, E. Brka, Near-perfect reconstruction low-complexity two-band IIR/FIR QMF banks with FIR phase-compensation filters. Sig. Process. 86, 171–181 (2005) Google Scholar
  16. F. Kurth, M. Clausen, Filter bank tree and M-band wavelet packet algorithms in audio signal processing. IEEE T. Sign. Process. 47(2), 549–554 (1999) Google Scholar
  17. B.G. Lee, A new algorithm to compute the discrete cosine transform. IEEE T. Acoust., Speech, Sign. Process. 32(6), 1243–1245 (1984) Google Scholar
  18. A.B. Lewis, JPEG Tutorial (The Computer Laboratory: Topics in Security: Forensic Signal Analysis, University of Cambridge, UK., 2010). https://www.cl.cam.ac.uk/teaching/1011/R08/jpeg/acs10-jpeg.pdf
  19. Y.P. Lin, P.P. Vaidyanathan, A Kaiser window approach for the design of prototype filters of cosine modulated filter banks. IEEE Sign. Process. Lett. 5(6), 132–134 (1998) Google Scholar
  20. H.W. Löllmann, P. Vary, Design of IIR QMF filter banks with near-perfect reconstruction and low complexity, in Proceedings of the IEEE International Conference Acoustics, Speech and Signal Processing, pp. 3521–3524 (2008) Google Scholar
  21. H.S. Malvar, Lapped transforms for efficient transform/subband coding. IEEE T. Acoust., Speech, Sign. Process. 38(6), 969–978 (1990) Google Scholar
  22. H.S. Malvar, D.H. Staelin, The LOT: Transform coding without blocking effects. IEEE T. Acoust., Speech, Sign. Process. 37(4), 553–559 (1989) Google Scholar
  23. J. Mau, Perfect reconstruction modulated filter banks: Fast algorithms and attractive new properties, in Proceedings of the IEEE International Conference Acoustics, Speech and Signal Processing, pp. 225–228 (1993) Google Scholar
  24. A. Mertins, Signal Analysis, Filter Banks, Time-Frequency Transforms and Applications (John Wiley, 1999) Google Scholar
  25. A. Mouffak, M.F. Belbachir, Noncausal forward/backward two-pass IIR digital filters in real time. Turk J. Elec. Eng. Comput. Sci. 20(5), 769–786 (2012) Google Scholar
  26. P.M. Naini, Digital watermarking using MATLAB. in Engineering Education and Research Using MATLAB, ed. by A. Assi (InTech, 2011) Chap. 20 Google Scholar
  27. T.Q. Nguyen, A Tutorial on Filter Banks and Wavelets (University of Wisconsin, ECE Department, 1995). http://www.cs.tau.ac.il/~amir1/WAVELET/PAPERS/nguyen95tutorial.pdf
  28. T.Q. Nguyen, P.P. Vaidyanathan, Two channel perfect reconstruction IR QMF structures which yield linear phase analysis and synthesis filters. IEEE T. Acoust., Speech, Sign. Process. 476–492 (1989) Google Scholar
  29. S. Oraintara, D. Trans, P.N. Heller, T.Q. Nguyen, Lattice structure for regular paraunitary linear-phase filterbanks and M-band orthogonal symmetric wavelets. IEEE T. Sign. Process. 49(11), 2659–2672 (2001) Google Scholar
  30. W. Pennebaker, J. Mitchell, JPEG: Still Image Data Compression Standard (Springer, 1992) Google Scholar
  31. C.I. Podilchuk, E.J. Delp, Digital watermarking algorithms and applications. IEEE Sign. Process. Mgz. 18(4), 33–46 (2001) Google Scholar
  32. R.L. Queiroz, T.Q. Nguyen, K.R. Rao, The GenLOT: Generalized linear-phase lapped orthogonal transform. IEEE T. Sign. Process. 44(3), 497–507 (1996) Google Scholar
  33. R.L. Queiroz, T.D. Tran, Lapped transforms for image compression, in Handbook on Transforms and Data Compression (CRC, 2000), pp. 1–64 Google Scholar
  34. C.M. Rader, The rise and fall of recursive digital filters. IEEE Sign. Process. Mgz. 46–49 (2006) Google Scholar
  35. K.R. Rao, P. Yip, Discrete Cosine Transform. Algorithms, Advantages, Applications (Academic Press, 1990) Google Scholar
  36. P.A. Regalia, S.K. Mitra, P.P. Vaidyanathan, The digital all-pass filter: A versatile signal processing building block. Proc. IEEE 76(1), 19–37 (1988) Google Scholar
  37. K. Sankar, Using Toeplitz Matrices in MATLAB (2007). http://www.dsplog.com/2007/04/21/using-toeplitz-matrices-in-matlab/
  38. I.W. Selesnick, Formulas for orthogonal IIR wavelet filters. IEEE T. Sign. Process. 46(4), 1138–1141 (1998) Google Scholar
  39. U. Sezen, Perfect reconstruction IIR digital filter banks supporting nonexpansive linear signal extensions. IEEE T. Sign. Process., 57(6), 2140–2150 (2009) Google Scholar
  40. U. Sezen, S.S. Lawson, Anticausal inverses for digital filter banks, in Proc. Eur. Conf. Circuit Theory Des. 1–229 to 1–232 (2001) Google Scholar
  41. M.J.T. Smith, T.P.III Barnwell, Exact reconstruction techniques for tree-structured subband coders. IEEE T. Acoust., Speech Sign. Process. 34(3), 434–441 (1986) Google Scholar
  42. A.K. Soman, P.P. Vaidyanathan, T.Q. Nguyen, Linear phase paraunitary filter banks: Theory, factorizations and design. IEEE T. Sign. Process. 41(12), 3480–3496 (1993) Google Scholar
  43. T.D. Tran, R.L. Queiroz, T.Q. Nguyen, Linear-phase perfect reconstruction filter bank: Lattice structure, design, and application in image coding. IEEE T. Sign. Process. 48(1), 133–147 (2000) Google Scholar
  44. P.P. Vaidyanathan, Multirate digital filters, filter banks, polyphase networks, and applications: A tutorial. Proc IEEE 78(1), 56–93 (1990) Google Scholar
  45. P.P. Vaidyanathan, Multirate Systems and Filter Banks (Prentice-Hall, 1992) Google Scholar
  46. P.P. Vaidyanathan, T. Chen, Structures for anticausal inverses and application in multirate filter banks. IEEE T. Sign. Process. 46(2), 507–514 (1998) Google Scholar
  47. P.P. Vaidyanathan, S.K. Mitra, Y. Neuvo, A new approach to the realization of low-sensitivity IIR digital filters. IEEE T. Acoust., Speech, Sign. Process. 34(2), 350–361 (1986) Google Scholar
  48. M. Vetterli, C. Herley, Wavelets and filter banks; theory and design. IEEE T. Sign. Process. 40(9), 2207–2232 (1992) Google Scholar
  49. M. Vetterli, J. Kovacevic, Wavelets and Subband Coding (Prentice Hall, 1995) Google Scholar
  50. A.B. Watson, Image compression using the discrete cosine transform. Math. J. 4(1), 81–88 (1994) Google Scholar
  51. Z. Xu, A. Makur, On the closeness of the space spanned by the lattice structures for a class of linear phase perfect reconstruction filter banks. Sign. Process. 90, 292–302 (2010) Google Scholar
  52. X. Zhang, T. Yoshikawa, Design of two-channel IIR Linear phase PR filter banks. Sign. Process. 72, 167–175 (1999) Google Scholar
  53. D. Zhou, A review of polyphase filter banks and their application. Technical report, Air Force Technical USA, 2006. AFRL-IF-RS-TR–277 Google Scholar

Author information

Authors and Affiliations

  1. Systems Engineering and Automatic Control, Universidad Complutense de Madrid, Madrid, Spain Jose Maria Giron-Sierra
  1. Jose Maria Giron-Sierra