In many RF/microwave communications systems, a bandpass filter with a flat group delay is demanded, in addition to its selectivity. One way to obtain this is by designing a self-equalized filter, in which the group delay equalizer is realized within the filter itself. In general, for a self-equalized filter, real and complex transmission zeros (TZ) are used to achieve both high selectivity and good linear phase performance; it has smaller structures, requires no cascading devices, but is relatively complex and difficult to design.1

In the cross-coupled filter synthesis, most of the publications focus on the coupling matrix synthesis. For example, Cameron proposed general coupling matrix synthesis methods for general Chebyshev filtering functions, based on an analytical technique.2,3 Numerical optimization was used in synthesizing coupling matrix of resonator filters.4-6 The analytical and numerical methods mentioned above can also be used to synthesize coupling matrix of self-equalized filters. However, the precondition of these methods is that the filter order and TZs must be prescribed. These methods did not discuss how to determine the filter order and TZs to meet the given filter specifications, although this is the foundation of the cross-coupled filter design. A self-equalized, pseudo-elliptical, filter based on the substrate integrated waveguide (SIW) technology was proposed,7 but how to determine the filter order and TZs was not discussed.

There are still few papers dealing with how to extract the filter order and TZs. The common solution for determining them is according to experiences and repeated tests. Although Ylldirim et al. used the circuit transformations to realize complex transmission zeros extraction and the cross-coupled filter synthesis with linear phase,8 the method involves complicated circuit transformations and is difficult to implement in some cases. H.T. Hsu et al. presented a design procedure for coupled-resonators cavity group-delay equalizers, whose TZs are obtained by optimization.1 However, the error function proposed for the optimization only contains the given filter group delay specifications, thus it cannot guarantee that the locations of TZs are optimum.

This article presents a simple and effective method to obtain the least filter order and optimum TZs to meet the given filter specifications, by minimizing an error function using GA. The proposed error function contains the given filter amplitude and group delay specifications. The proposed method eliminates the need for circuit model and only depends on the given filter specifications. The article solves the first problem of coupling matrix synthesis of resonator filters. Once the filter order and the locations of TZs are determined, the coupling matrix of self-equalized filters can be obtained by analytical methods2,3 or numerical optimization methods.4-6 The next step requires the choice of proper resonator types, for example SIW7 or stepped-impedance resonators,9 to realize the self-equalized filter design.

Theory

For any two-port lossless filter network, the transfer and reflection functions may be expressed as a ratio of two polynomials.3

Where Ω is the real normalized angular frequency variable, the roots of P and F correspond to the filter's TZs and reflection zeros, respectively,

RL is the prescribed return loss in decibels, and εR is unity if m3

According to the group delay in the Ω domain,4 the actual value of the group delay in the physical frequency f domain is derived as

The frequency transformation from Ω to f can be expressed as follows,

where FBW is the fractional bandwidth, f0 is the center frequency of the bandpass filter. Equation 2 may now be expressed using Equations 1 and 3 as follows,

Once the filter order and the finite TZs are determined in the Ω domain, E(Ω), F(Ω) and P(Ω) are determined, based on Cameron's recursive technique.2 Both the amplitude S21(f) and group delay τ(f) response of the filter, which has a center frequency f0 and a fractional bandwidth FBW, are determined in the f domain, since Ω in Equations 1 and 4 is calculated by Equation 3. So, determining the filter order and TZs is the key problem of filter synthesis.

Usually, self-equalized filter specifications are listed as follows:
Center frequency: f0
Fractional bandwidth: FBW
Passband return loss: RL (dB)
Group delay variation: <Δτ (ns), when fp1 ≤ f ≤ fp2

Stopband rejection: Las ≥ Ls (dB), when f > fs1 or/and f s2

So, two functions are constructed as:

The function φ1 denotes that the filter has the highest stopband rejection, when φ1 reaches its minimum. The function φ2 denotes that the filter has the flattest group delay response, when φ2 reaches its minimum. The proposed error function for GA is as follows:

where C is the weighted constant determined by C = 10 -Ls/10/Δτ. Note that C is the key parameter that determines a correct solution. In order to meet the given filter specifications, φ must satisfy φ ≤ 0. An error function similar to Equation 7, for more than two requirements in given filter specifications, can be constructed.

For self-equalized bandpass filters, TZs (Ωn) on Ω-axis are used to increase selectivity, and complex pairs of TZs (Ti = Ωai ± jσai), which are called equalization zeros, are used to equalize in-band group delay. In general, the better the group delay equalization, the worse the selectivity performance for a filter. Hence, the group delay and the selectivity can be balanced in the filter designs. Ωn, Ωai and σai making up finite TZs will be used as optimized variables in GA. In this article, the GA toolbox for MATLAB, provided by the University of Sheffield, is chosen to solve the global minimum of the error function. The range of parameters to be optimized can be set in this toolbox. The toolbox is easy to download from the Internet.10 The search for TZs to minimize Equation 7 is done by increasing the filter order and the number of the TZs gradually, using GA, until φ < 0, and then the least filter order and the optimum TZs can be obtained to meet filter specifications. This method is easy to determine the TZs and filter order of a self-equalized filter with symmetric or asymmetric response.

Examples

Filter 1

A self-equalized filter (filter 1) is required to meet the following specifications:

  • Center frequency: 35 GHz
  • Bandwidth: 1.4 GHz (fractional bandwidth = 4 percent)
  • Passband return loss: 20 dB
  • Group delay variation: < 0.025 ns, when 34.65 ≤ f ≤ 35.35 GHz (within 50 percent bandwidth)
  • Stopband rejection Las > 35 dB, when f > 36.08 GHz and f < 33.92 GHz

Figure 1 Comparison of the filter responses computed from this article and reference 7 (a) amplitude and (b) group delay.

These specifications were taken from Chen and Wu's article7 to provide a direct comparison. In the example, the real TZs (Ωn) is symmetric with respect to zero on Ω-axis, since the required amplitude specification is symmetric with respect to the center frequency. The least filter order N=6 and four TZs Ω1,2 = ± 1.6044 and T 1,2 = 0.0317 ± 1.0441j are obtained according to the proposed method, and the value of the error function is -1.96×10-4. In Chen and Wu's article, the filter order N = 6, four TZs Ω1,2 = ± 1.6060 and T1,2 = ± 1.0356j are given, and the value of error function is equal to -1.47×10-6, calculated by substituting TZs and filter order obtained from Chen's into Equation 7. The amplitude and group delay responses of filter 1 are computed according to TZs and filter order given in this article and in Chen's paper, using Equation 1, and are shown in Figure 1. A comparison of the values of the error function and the filter responses shows that the locations of TZs given in this article are much better, which validate the proposed method. The detailed comparison of the group delay in the passband is zoomed in, demonstrating the equal-ripple group delay of the filter synthesized by the proposed method.

Filter 2

The specifications for filter 2 are as follows:

  • Center frequency: 1960 MHz
  • Bandwidth: 60 MHz (fractional bandwidth = 3.06 percent)
  • Passband return loss: 20 dB
  • Group delay variation: < 0.5 ns, when 1945 ≤ f ≤ 1975 MHz (within 50 percent bandwidth)
  • Stopband rejection: Las > 25 dB, when f < 1920 MHz.
    The least filter order N = 5 and three TZs Ω1 = -1.4569 and T1,2 = 0.2569 ± 1.1420j can be obtained, according to the proposed method, to meet the specifications of filter 2, and the value of the error function is -4.73×10-4. The realizable coupling coefficient matrix and external quality factor obtained by Cameron's method 2 are given in Equation 8, since the filter order and TZs have been determined by the proposed method.

The amplitude and group delay response are shown in Figure 2, which are obtained from the coupling coefficient matrix in Equation 8 and the transfer and reflection functions in Equation 1, respectively. The difference between the two results is not visible, which confirms the validity of the proposed method.

Figure 2 Synthesized responses of filter 2 from equations 1 and 8 (a) amplitude and (b) group delay.

Filter 3

The specifications for filter 3 are as follows:

  • Center frequency: 1800 MHz
  • Bandwidth: 60 MHz
  • Passband return loss: 20 dB
  • Group delay variation: < 0.35 ns over the central 60 percent of the passband
  • Rejection at f0 ± 64 MHz: > 15 dB

The least filter order N = 4 and T1,2 = ± 1.6575j are obtained according to the proposed method to meet the specifications of filter 3, and the value of the error function is -0.036. The coupling coefficients and external quality factor obtained by Amari's method4 are:

After the coupling coefficients are determined, the next step requires the choice of proper resonator types to complete the filter design. In this design, a half-wavelength resonator was chosen.

A full-wave simulator IE3D has been used to extract the parameters in Equation 9. In general, the coupling coefficient Kij between two resonators i and j, and the external quality factor are evaluated as11

Figure 3 Physical dimensions (a) and photograph (b) of filter 3.

Where f1 and f2 are the lower and upper resonant peaks in the transmission response, respectively, and f0 and Δf3-dB represent the resonant frequency and the 3 dB bandwidth of the input and output resonators, when externally excited. The filter is designed to be fabricated on a Rogers RT/duroid 5880 substrate, with a relative dielectric constant of 2.2, a thickness of 0.508 mm, and a loss tangent of 0.0009. A photograph and the physical dimensions of filter 3 are shown in Figure 3. By using open-loop and meandering half-wavelength resonators configurations, unwanted couplings can be reduced effectively.

The measured and simulated results of the filter are shown in Figure 4. The measured results are a bit worse than the simulated results because of fabrication errors and the influence of the transition between sub-miniature version A (SMA) connectors and microstrip. This example illustrates the general design procedure for self-equalized cross-coupled resonator bandpass filters whose TZs are obtained by the proposed method.

Conclusion

This article has presented a simple and efficient method to determine the least filter order and optimum TZs of self-equalized filters, to meet the given amplitude and group delay specifications using GA optimization method. To prove the validity of the proposed method, the filter order and TZs of three self-equalized filters have been determined. A fourth-order self-equalized filter has been designed and fabricated, which illustrates the general design procedure for self-equalized cross-coupled resonator bandpass filters. The proposed method has solved the first problem of coupling matrix synthesis of self-equalized filters.

Figure 4 Measured and simulated frequency responses of filter 3 (a) amplitude and (b) group delay.

References

  1. H.T. Hsu, H.W. Yao, K.A. Zaki and A.E. Atia, "Synthesis of Coupled-resonators Group-delay Equalizers," IEEE Transactions on Microwave Theory and Technique, Vol. 50, No. 8, August 2002, pp. 1960–1968.
  2. R.J. Cameron, "General Coupling Matrix Synthesis Methods for Chebychev Filtering Functions," IEEE Transactions on Microwave Theory and Technique, Vol. 47, No. 4, April 1999, pp. 433–442.
  3. R.J. Cameron, "Advanced Coupling Matrix Techniques for Microwave," IEEE Transactions on Microwave Theory and Technique, Vol. 51, No. 1, January 2003, pp. 1–10.
  4. S. Amari, "Synthesis of Cross-coupled Resonator Filters Using an Analytical Gradient-based Optimization Technique," IEEE Transactions on Microwave Theory and Technique, Vol. 48, No. 9, September 2000, pp. 1559–1564.
  5. M. Uhm, S. Nam and J. Kim, "Synthesis of Resonator Filters with Arbitrary Topology Using Hybrid Method," IEEE Transactions on Microwave Theory and Technique, Vol. 55, No. 10, October 2007, pp. 2157–2167.
  6. G.L. Nicholson and M.J. Lancaster, "Coupling Matrix Synthesis of Cross-coupled Microwave Filters Using a Hybrid Optimization Algorithm," IET Microwaves, Antennas Propagation, Vol. 3, No. 6, September 2009, pp. 950–958.
  7. X.P. Chen and K. Wu, "Self-equalized Pseudo-elliptical Filter Made of Substrate Integrated Waveguide," Electronics Letters, Vol. 45, No. 2, January 2009, pp. 112–113.
  8. N. Ylldirim, O.A. Sen, Y. Sen, M. Karaaslan and D. Pelz, "A Revision of Cascade Synthesis Theory Covering Cross-coupled Filters," IEEE Transactions on Microwave Theory and Technique, Vol. 50, No. 6, June 2002, pp. 1536–1543.
  9. M. Sagawa, M. Makimoto and S. Yamashita, "Geometrical Structures and Fundamental Characteristics of Microwave Stepped-impedance Resonators," IEEE Transactions on Microwave Theory and Technique, Vol. 45, No. 7, July 1997, pp. 1078–1085.
  10. GA Toolbox, http://www.shef.ac.uk/acse/research/ecrg/getgat.html.
  11. J.S. Hong and M.J. Lancaster, "Microstrip Filters for RF/microwave Applications," Wiley, New York, 2001, pp. 235–265.

Rui Wang received his bachelor's degree in physics and electronics from Ludong University, Yanta, China, in 2004, and his master's degrees in physics and electronics from Xidian University in 2007. From 2007 to 2009, he was a Research Engineer with China Air to Air Missile Academy, where he was involved with the design of antennas and components for transmitters and receivers. Since 2009, he has been working toward Ph.D. degrees in physics and electronics at the University of Electronic Science and Technology of China (UESTC). His current research interests include computer-aided millimeter-wave circuit, passive component design, electromagnetic field theories and numerical analysis.

Jun Xu received his bachelor's degree in 1984 and his master's degrees in 1990 in electronics and engineering from the University of Electronic Science and Technology of China (UESTC), Chengdu, China. Since 1984, he has been with the School of Physics and Electronics at the UESTC, China, where he is currently a Professor. His research interests include millimeter-wave circuits and systems and transmitters, and receivers for millimeter-wave radar system.