General Closed-Form Transfer Function Expressions for Fast Filter Bank

The existing literature focuses on the applications of fast filter bank due to its excellent frequency responses with low complexity. However, the topic is not addressed related to the general transfer function expressions of the corresponding subfilters for a specific channel. To do this, in this paper, general closed-form transfer function expressions for fast filter bank are derived. Firstly, the cascaded structure of fast filter bank is modelled by a binary tree, with which the index of the subfilter at each stage within the channel can be determined. Then the transfer functions for the two outputs of a subfilter are expressed in a unified form. Finally, the general closed-form transfer functions for the channel and its corresponding subfilters are obtained by variables replacement if the prototype lowpass filters for the stages are given. Analytical results and simulations verify the general expressions. With such closed-form expressions lend themselves easily to analysis and direct computation of the transfer functions and the frequency responses without the structure graph.