Research on Bandpass Filter Based on Mathematical Optimization Method

Microwave filters are an important part of wireless communication transceiver circuits. They are responsible for signal selection at the front end of transceiver and are widely used in various wireless systems. The research on their design methods has always been one of the focus issues in microwave academic research. Usually, filters can be designed using traditional circuit synthesis methods, including image parameter method, network synthesis method, insertion loss method, etc. In recent years, with the increasing development of wireless communication technology and the continuous reduction of microwave filter area, it is difficult to meet the design requirements of modern new filters with more complex structures using traditional design methods alone. Therefore, proposing new design methods has become one of the key issues in technological development and application.
Based on the above background, this paper proposes a mathematical optimization design method for filter design. Compared with the traditional circuit synthesis method, this optimization design method can set up target functions for multiple frequency points and achieve approximation of multiple targets within a frequency range, thereby effectively grasping the overall response of the filter and achieving the purpose of accurate design and improving performance. This paper uses this method to design an equi-ripple inductor coupled bandpass filter to verify its superiority in controlling bandwidth and improving overall performance.

1 Design principle of mathematical optimization method
Mathematical optimization method is a method for finding extreme values, that is, making the objective function reach the extreme value under a set of constraints. Commonly used mathematical optimization methods include gradient method, Newton method, annealing algorithm and genetic algorithm.
In the process of filter optimization design, the optimization frequency point is first obtained, and then the objective function K is constructed, and its minimum value at the optimization frequency point is taken as the target, and the optimal solution is obtained by using the optimization algorithm.
1.1 Solving the optimization frequency point
The equiripple bandpass filter is designed based on Chebyshev polynomials. Since the frequency response of the bandpass filter is obtained by transforming the frequency response of the low-pass prototype, the low-pass filter is considered first.
The filter function PLR of the generalized Chebyshev low-pass prototype is a rational function of frequency ω, and PLR is determined by its zeros, poles and a constant. If the insertion loss response of the N-order low-pass prototype is set by Chebyshev polynomials, the filter function PLR can be obtained as follows:

From equations (2) and (3), it can be obtained that the zeros of S11 and S21 are ωzi and ωpi (i is determined by N).
Assume that the final goal is to achieve a bandpass filter with a center frequency of ω0. Now convert the low-pass response to a bandpass response according to formula (4).

In the formula: △, ω0 are the relative bandwidth and center frequency of the bandpass filter respectively; ωi' is the frequency point of the low-pass response: ωi is the corresponding frequency point of ωi' after the low-pass response is converted to a bandpass response.
From formula (4),

we get the zero points ωzi and ωpi of S11 and S21 in the corresponding bandpass filter, which are the optimized frequency points to be solved.
1.2 Construction of the objective function
Taking into account the problems of dielectric loss, component loss, and dimensional error in practice, the functional relationships (1), (2), and (3) under the ideal conditions mentioned in 1.1 are actually unrealizable. Since the zero point of PLR is the zero point of the reflection function S11, and its pole is the zero point of the transmission function S21, the zero points of S11 and S21 can be selected to construct the objective function. As for the constant k, it can be calculated from the reflection coefficient at ω=±1. Based on this, the following objective function is made:

This paper proposes to use the gradient optimization algorithm to approximate the objective function. First, the gradient of the objective function K is obtained. In order to make the optimization process more effective, the gradient of the objective function is used to search for its extreme value. This gradient-based optimization method converges very quickly, and there will be no situation where the optimization process cannot converge or converges to a local extreme point, so that the objective function can be approximated quickly and accurately.
2 Simulation and comparison of filters
To verify the effectiveness of this method, a third-order 0.5 dB equi-ripple inductor-coupled bandpass filter with a center frequency of 2 GHz and an equi-ripple bandwidth of 30% is designed using the insertion loss method.
The transmission line model of the equi-ripple inductor-coupled bandpass filter is shown in Figure 1, where the characteristic impedance of all transmission lines is Z0, which is generally selected as 50 Ω, θi is the electrical length of the i-th transmission line, jXi is the impedance of the corresponding i-th inductor, and the two transmission lines at both ends of the model are lead lines.

The transmission line model is actually equivalent to a structure of an impedance inverter and a half-wavelength resonator in series, based on which the electrical length and inductance of each transmission line can be calculated. The microstrip implementation layout of the third-order 0.5 dB equi-ripple inductor coupled bandpass filter obtained using the insertion loss method is shown in Figure 2.

The size of each microstrip line is:

On this basis, mathematical optimization is performed, and the optimization target is min K. According to the results of the optimization calculation, the sizes of each transmission line in the filter are as follows (corresponding to the parameters in Figure 2):

The simulation results of transmission functions S11 and S12 are shown in Figure 3.

As shown in Figure 3(a), the bandwidth of the passband in the simulation results of the insertion loss method is about 26%, and after the mathematical optimization method is used on this basis, the passband bandwidth increases to the set 30%. In addition, Figure 3(b) shows that the optimized result improves the original equiripple characteristics, is closer to the ideal response, and the filter performance is improved.

3 Conclusion
This paper proposes a mathematical optimization design method for the design of filters. Theoretical analysis and simulation results show that this method has great advantages over the traditional insertion loss method. This method can quickly and accurately approach the target by setting the objective function and constraints, approach the ideal response, and greatly improve the filter performance. Therefore, this method can be widely used in the design process of filters.