Abstract-Two digital filters ( ) and ( ) are said to be biorthogonal partners of each other if their cascade ( ) ( ) satisfies the Nyquist or zero-crossing property. Biorthogonal partners arise in many different contexts such as filterbank theory, exact and least squares digital interpolation, and multiresolution theory. They also play a central role in the theory of equalization, especially, fractionally spaced equalizers in digital communications. In this paper, we first develop several theoretical properties of biorthogonal partners. We also develop conditions for the existence of biorthogonal partners and FIR biorthogonal pairs and establish the connections to the Riesz basis property. We then explain how these results play a role in many of the above-mentioned applications.

Multiple Input Multiple Output (MIMO) biorthogonal partners arise in many different contexts, one of them being multiwavelet theory. They also play a central role in the theory of MIMO channel equalization, especially with fractionally spaced equalizers. In this paper we first derive some theoretical properties of MIMO biorthogonal partners. We develop conditions for the existence of MIMO biorthogonal partners and conditions under which FIR solutions are possible. In the process of constructing FIR MIMO biorthogonal partners we exploit the non-uniqueness of the solution. This will lead to the design of flexible fractionally spaced MIMO zero-forcing equalizers. The additional flexibility in design makes these equalizers more robust to channel noise. Finally, other situations where MIMO biorthogonal partners occur will also be considered, such as prefiltering in multiwavelet theory and deriving the vector version of the least squares signal projection problem.

No abstract

Abstract-Channel equalization is an important step in most applications of digital communications. In this paper we consider the equalization of Multiple Input Multiple Output (MIMO) channels. To that end we derive the theory of MIMO biorthogonal partners, a concept that has already been introduced (only in the scalar case) by the authors. We develop conditions for the existence of an FIR MIMO biorthogonal partners and describe their application in the MIMO channel equalization. We also show that it is possible to exploit the non-uniqueness of FIR MIMO biorthogonal partners in order to design flexible fractionally spaced MIMO equalizers that will be more robust to the channel noise.

The concept of biorthogonal partners has been introduced recently by the authors. The work presented in this paper is an extension of some of these results to the case where the upsampling and downsampling ratios are not integers but rational numbers. Hence the name fractional biorthogonal partners. The conditions for the existence of stable and of FIR fractional biorthogonal partners are derived. This result gives rise to an all-FIR spline interpolation technique with the minimum amount of required oversampling. This technique is illustrated by an interpolation example.