Design of Modified Second-Order Frequency Transformations Based Variable Digital Filters with Large Cutoff Frequency Range and Improved Transition Band Characteristics

The frequency transformation based filters (FT filters) provide an absolute control over the cutoff frequency. However, the cutoff frequency range ( Ω {{crm range}}) of the FT filters is limited. The second-order frequency transformations combined with coefficient decimation technique based filter (FTCDM filter) has wider Ω {{crm range}} compared with the FT filter; however, the ratio of transition bandwidth of the transformed filter to that of the prototype filter, tbw rm FT} tbw rm mod} , is large over a significant portion of Ω {{crm range}}. In this paper, we propose a novel idea of relaxing the one-to-one mapping condition between the frequency variables, to overcome the issue of limited Ω {{crm range}} for tbw rm FT} le textit {tbw}rm mod}. In the proposed modified second-order frequency transformation based filter (MSFT filter), we relax the one-to-one mapping condition between the frequency variables and use low-pass to high-pass transformation on the prototype filter to achieve wider Ω {{crm range}} with tbw rm FT} le textit {tbw}rm mod}. Design example shows that the MSFT filter provides 3 and 1.22 times wider Ω {{crm range}} compared to FT and FTCDM filters, respectively.