Degrees of Freedom of a Communication Channel: Using DOF Singular Values

A fundamental problem in any communication system is: given a communication channel between a transmitter and a receiver, how many ¿independent¿ signals can be exchanged between them? Arbitrary communication channels that can be described by linear compact channel operators mapping between normed spaces are examined in this paper. The (well-known) notions of degrees of freedom (DOF) at level ¿ and essential dimension of such channels are developed in this general setting. We argue that the DOF at level ¿ and the essential dimension fundamentally limit the number of independent signals that can be exchanged between the transmitter and the receiver. We also generalize the concept of singular values of compact operators to be applicable to compact operators defined on arbitrary normed spaces which do not necessarily carry a Hilbert space structure. We show how these generalized singular values, which we call DOF singular values, can be used to calculate the DOF at level ¿ and the essential dimension of compact operators that describe communication channels. We describe physically realistic channels that require such general channel models.

[1]  Helmut Bölcskei,et al.  On the capacity of OFDM-based spatial multiplexing systems , 2002, IEEE Trans. Commun..

[2]  P. K. Chaturvedi,et al.  Communication Systems , 2002, IFIP — The International Federation for Information Processing.

[3]  D. A. Bell,et al.  Information Theory and Reliable Communication , 1969 .

[4]  Minyue Fu,et al.  Wireless communication systems with-spatial diversity: a volumetric model , 2006, IEEE Transactions on Wireless Communications.

[5]  Antonia Maria Tulino,et al.  Random Matrix Theory and Wireless Communications , 2004, Found. Trends Commun. Inf. Theory.

[6]  E. Kreyszig.  Introductory Functional Analysis With Applications , 1978 .

[7]  D. Miller,et al.  Communicating with waves between volumes: evaluating orthogonal spatial channels and limits on coupling strengths. , 2000, Applied optics.

[8]  Ram Somaraju,et al.  Essential dimension and degrees of freedom for spatial waveform channels , 2008 .

[9]  M. Migliore.  On the role of the number of degrees of freedom of the field in MIMO channels , 2006, IEEE Transactions on Antennas and Propagation.

[10]  Rodney A. Kennedy,et al.  Intrinsic Limits of Dimensionality and Richness in Random Multipath Fields , 2007, IEEE Transactions on Signal Processing.

[11]  R. Janaswamy,et al.  Electromagnetic Degrees of Freedom in 2-D Scattering Environments , 2006, IEEE Transactions on Antennas and Propagation.

[12]  R. Gallager.  Information Theory and Reliable Communication , 1968 .

[13]  G. Franceschetti,et al.  On the degrees of freedom of scattered fields , 1989 .

[14]  Robert W. Brodersen,et al.  Degrees of freedom in multiple-antenna channels: a signal space approach , 2005, IEEE Transactions on Information Theory.

[15]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[16]  Sang Joon Kim,et al.  A Mathematical Theory of Communication , 2006 .

[17]  Giorgio Taricco,et al.  Transmission and Reception with Multiple Antennas: Theoretical Foundations , 2004, Found. Trends Commun. Inf. Theory.

[18]  Paul D. Alexander,et al.  Random Sequence Multisets for Synchronous Code-Division Multiple-Access Channels , 1998, IEEE Trans. Inf. Theory.

[19]  Sergio Verdú,et al.  Fifty Years of Shannon Theory , 1998, IEEE Trans. Inf. Theory.

[20]  D. Slepian,et al.  On bandwidth , 1976, Proceedings of the IEEE.

[21]  H. Pollak,et al.  Prolate spheroidal wave functions, fourier analysis and uncertainty — III: The dimension of the space of essentially time- and band-limited signals , 1962 .

[22]  G. Franceschetti,et al.  On the spatial bandwidth of scattered fields , 1987 .