SAW Filter Optimization Techniques

Abstract

This lecture focuses on SAW filter optimization techniques used in IF SAW filter design. It addresses both optimum and suboptimum synthesis of SAW bandpass filters with prescribed magnitude and phase responses. In general, SAW filters may exhibit arbitrary magnitude characteristics as well as linear-, nonlinear-, or minimum-phase responses.

Linear-Phase SAW filters

First, the lecture considers the design of linear-phase SAW filters. It then introduces two different approaches to the optimum synthesis problem. In the first approach, a SAW filter consists of two linear-phase interdigital transducers (IDTs). The frequency response of one transducer is specified a priori, while the other is optimized to achieve the desired overall magnitude response.

To approximate the target response, the method applies a weighted Chebyshev (minimax) criterion. The approach is general and does not impose constraints on the magnitude shape, which may be symmetrical, asymmetrical, or multi-passband. The lecture formulates the optimum minimax design problem rigorously. As a result, the method provides the best possible fit to the design target and yields a minimum-length SAW filter under the given constraints.

Furthermore, the lecture shows how to transform the original approximation problem into an auxiliary one by properly predistorting the desired magnitude and weight functions. This auxiliary problem can then be solved using standard Chebyshev approximation techniques, such as linear programming (simplex method) or the Remez exchange algorithm (e.g., McClellan’s Computer Program for Designing Optimum FIR Linear Phase Digital Filters). Finally, the method can additionally incorporate the element factor and/or the multistrip frequency response.

The major drawback of optimum design methods (for example, McClellan’s program) is the long computational time and numerical instability when the number of optimized variables becomes large. This situation is typical for narrowband SAW filters, where the number of variables increases inversely with the fractional bandwidth. However, for band-limited responses, the number of optimized variables can be significantly reduced by applying the sampling theorem in either the frequency or time domain.

A proprietary suboptimum synthesis technique based on McClellan’s program is proposed to address this issue. In this approach, the number of optimized variables is reduced without significant loss of approximation accuracy. The method achieves this by factorizing the optimized function a priori in the Z-transform domain. In particular, most stopband equidistant zeros are assigned analytically in closed form to a predefined function, thereby reducing the order of the optimized function. As a result, both computation time and memory requirements are substantially reduced compared to the optimum design.

The lecture discusses both the theoretical background and practical aspects of the suboptimum synthesis method and illustrates them with design examples. For linear-phase SAW filter design, three optimization techniques are compared:

1. Remez Exchange Algorithm (REA)
2. Linear Programming (LP)
3. Iterative Weighted Least Squares (IWLS)

All three methods produce nearly identical suboptimum solutions. However, they differ in computational cost and implementation complexity. In particular, the proprietary IWLS algorithm developed by the author is the fastest and easiest to implement. Moreover, many numerical packages, including MATLAB®, provide least-squares routines, which form the core of the IWLS method.

An alternative approach to SAW filter design is factorizational synthesis. In this method, the design starts from an overall SAW filter response optimized without a priori constraints on the input and output transducers. Either optimum or suboptimum techniques can be used at this stage.

Next, the trigonometric polynomial representation of the response is converted into an algebraic polynomial using the Z-transform. The roots (zeros) are then computed using a high-order solver, after which they are distributed between the input and output transducers. Finally, the tap weights of each transducer are recovered from their respective Z-transform representations.

In the general case, this procedure results in two apodized SAW transducers, and a multistrip coupler (MSC) is required for acoustic coupling in a dual-track configuration. However, for narrowband SAW filters, one transducer can be withdrawal-weighted (or polarity-weighted), enabling a single-track topology.

Nonlinear-Phase SAW filters

The lecture also considers the most general optimization problem for SAW filter synthesis with arbitrary magnitude and phase responses. It proposes design schemes based on the separate optimization of the real and imaginary parts of the complex-valued frequency response. Compared to nonlinear programming (NLP), these schemes are faster and more efficient.

To further improve approximation accuracy, the author develops iterative design algorithms in which the tolerance field is dynamically updated at each iteration based on the results of the previous step.

An alternative approach based on the Euclidean metric is also presented. This method applies Chebyshev approximation to a complex-valued function and supports both single-step and iterative optimization. Unlike REA or LP methods, the proprietary IWLS algorithm can be naturally extended to the complex domain with dynamically updated weighting functions. All optimization techniques (REA, LP, NLP, IWLS) have been implemented by the author in MATLAB.

The lecture includes design examples for each method and concludes with a live demonstration of the optimization software in practical SAW filter design.

Contents

1. Introduction

2. Optimum design of SAW linear-phase bandpass filters

3. Chebyshev approximation and its properties

4. Optimization methods

4.1 Linear programming (LP)

4.2 Remez exchange algorithm (REA)

4.3 Iterative Weighted Least Squares (IWLS)

5. Suboptimum synthesis of SAW filters

5.1 Frequency sampling technique and LP optimization

5.2 Optimum and suboptimum synthesis using REA

6. Factorizational synthesis of SAW filters

7. Design of SAW filters with prescribed responses

7.1 Linearization schemes

7.2 Iterative techniques

7.3 Chebyshev approximation of complex-valued functions

7.4 WLS approximation in the complex domain

8. Design examples

9. Conclusions