Transversal filter synthesis based on frames of translates

In this paper we present a new methodology for transversal filter synthesis based on frames of integer-translates. The method based on frame theory is an alternative approach to traditional transversal filter synthesis and very suitable for the tapped-delay structure. Furthermore we provide necessary and sufficient conditions for {f(.-k)}(k is an element of Z) to form a Riesz basis or frame sequence for l(2)(Z). If the translated versions of input signal form a Riesz basis, the bi-directional Levinson algorithm can be used to approximate filter coefficients directly, whose computational cost is modest and storage requirement is less. Besides, we extend finite transversal filter to infinite transversal filter which can be formulated exactly by systems of integer-translates and regard the former as a projection of the latter. Finally, we present the geometrical nature of finite transversal filter synthesis based on Riesz bases of translates.