Unified Theory: Multirate Transformations

  • Gianfranco Cariolaro


Multirate transformations, better known as multirate systems, operate on discrete signals that have different rates. A single discrete transformation (tf) is characterized by two rates, the input and the output rate, but in a combination of tfs we may find several rates. Typical components of multirate tfs belong to the class of QIL tfs, which include up-samplers, down-samplers, interpolators, decimators and fractional interpolators. Sometimes, exponential modulators, which belong to the broader class of PIL tfs, are considered. The peculiarity of multirate tfs lies in certain properties which allow several alternative architectures with interesting consequences for applications. In this chapter, multirate systems are developed in the signal-domain, where the Unified Signal Theory is particularly useful for the relevance it gives to the domains and consequently to the rates. The signal-domain approach is atypical in the literature where the z-domain-approach is commonly used. Another difference with the literature is in multidimensional multirate systems where we use a representation-independent formulation of lattice operations, whereas other authors make use of heavy matrix manipulations.


Orthogonal Frequency Division Multiplex Impulse Response Finite Group Orthogonal Frequency Division Multiplex System Parallel Architecture 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    M.G. Bellanger, J.L. Daguet, TDM–FDM transmultiplexer: digital polyphase and FFT. IEEE Trans. Commun. COM-22, 1199–1205 (1974) CrossRefGoogle Scholar
  2. 2.
    M.G. Bellanger, G. Bonnerot, M. Coudress, Digital filtering by polyphase network: application to sample rate alteration and filter banks. IEEE Trans. Acoust. Speech Signal Process. ASSP-24, 109–114 (1976) CrossRefGoogle Scholar
  3. 3.
    J.A.C. Bingham, Multicarrier modulator for data transmission: an idea whose time has come. IEEE Commun. Mag. 5–14 (1990) Google Scholar
  4. 4.
    C.S. Burrus, T.W. Parks, DFT/FFT and Convolution Algorithms Theory and Implementation (Wiley, New York, 1985) Google Scholar
  5. 5.
    G. Cariolaro, Theory of multidimensional periodically invariant linear system. CESP Report, Dept. Information Engineering, University of Padova, December 2003 Google Scholar
  6. 6.
    G. Cariolaro, A.M. Cipriano, F. De Pellegrini, New Noble Identities for multidimensional multirate linear systems based on exponential modulators, in Proceedings of SPACS 2003, Osaka JP, December 2003 Google Scholar
  7. 7.
    G. Cariolaro, V. Cellini, G. Donà, Theoretic group approach to multidimensional orthogonal frequency division multiplexing, in ISPACS 2003, Awaji Island, Japan, December 2003 Google Scholar
  8. 8.
    G. Cariolaro, P. Kraniauskas, L. Vangelista, A novel general formulation of up/downsampling commutativity. IEEE Trans. Signal Process. 53, 2124–2134 (2005) MathSciNetCrossRefGoogle Scholar
  9. 9.
    T. Chen, P. P Vaidyanathan, The role of integer matrices in multidimensional multirate systems. IEEE Trans. Signal Process. SP-41, 1035–1047 (1993) CrossRefGoogle Scholar
  10. 10.
    J. Chow, J. Tu, J. Cioffi, A discrete multitone transceiver system for HDSL applications. IEEE J. Sel. Areas Commun. 9, 895–908 (1991) CrossRefGoogle Scholar
  11. 11.
    S. Coulombe, E. Dubois, Non-uniform perfect reconstruction filter banks over lattices with applications to transmultiplexers. IEEE Trans. Signal Process. 47 (1999) Google Scholar
  12. 12.
    S. Darlington, On digital single-sideband modulators. IEEE Trans. Circuit Theory CT-17, 409–414 (1970) CrossRefGoogle Scholar
  13. 13.
    J. Kovac̆ević, M. Vetterli, The commutativity of up/downsampling in two dimensions. IEEE Trans. Inf. Theory 37(3) (1991) Google Scholar
  14. 14.
    P.P. Vaidyanathan, Multirate digital filters, filter banks, polyphase networks, and applications: a tutorial. Proc. IEEE 78, 56–93 (1990) CrossRefGoogle Scholar
  15. 15.
    P.P. Vaidyanathan, Multirate Systems and Filter Banks (Prentice Hall, Englewood Cliffs, 1993) MATHGoogle Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

There are no affiliations available

Personalised recommendations