Factorizational Synthesis of SAW Filters
Abstract
The lecture discusses properties of Z-transform roots. Furthermore, the roots are classified with regard to their symmetry, invertibility and phase. They can be grouped according to their location with respect to the unit circle. These groups contain a single zero (monozero) on the real axis, two mirror zeros (bi-zero) on or off the unit circle and four inverse and mirror zeros (quadroplet, or quadro-zero) off the unit circle. Passband and stopband magnitude and phase responses of each group and their contributions to the SAW filter response are analyzed.
Linear- and Nonlinear-Phase SAW Filter Synthesis
Several roots separation schemes and algorithms are discussed. For LP design, both IDTs may have identical NLP responses, which cancel each other due to complex conjugation in the overall SAW filter response. This phase cancellation ensures the LP SAW filter response within the entire frequency range.
For narrowband SAW filters, apodized LP or NLP IDTs can be approximated by withdrawal-weighting (or polarity-weighting). This allows to convert dual-track SAW filter topology with multistrip coupler (MSC) into conventional single track (in-line) one. Given the design specifications, the algorithm results in the minimum length SAW filters which are typically 20- 30 % shorter if compared to the conventional designs techniques (for example, with two identical apodized IDTs).
Minimum-Phase and Quasi-Minimum-Phase SAW Filters
The algorithm can be adopted to design optimum (or suboptimum) MP SAW filters from the LP prototype. The only additional step is inversion of the outside Z-transform roots inside the unit circle. Such a MP design minimizes SAW filter time delay if compared to the LP SAW filter that could be useful in some applications.
The length of MP SAW filters may be further reduced by applying more sophisticated optimum design procedures providing “minimum-minimorum” length SAW filters for prescribed magnitude shape specifications. Two different MP synthesis algorithms based on the LP prototype are considered which are similar to the MP design of non-recursive finite impulse response (FIR) digital filters. A general principle is moving LP prototype stopband roots off the unit circle. The first method is known in digital filtering as Hermann-Shuessler procedure, where the optimum equiripple LP response is offset in such a way that stopband zero pairs merge into the double zeros. Alternatively, a direct optimal synthesis of the positive real-valued LP prototype can be applied. In both methods, the desired MP response is a square root of this LP prototype response.