1 Introduction

The traditional radio frequency (RF) transceiver system consists of antenna, filter, power amplifier, low noise amplifier, and other devices, which often work under a single communication standard. If multiple communication standards run at the same time, it requires multiple independenttransceiver systems to form a parallel working system which will be larger in size, high in power consumption and high in cost, and has been unable to meet the application needs of wireless sensor network nodes in the era of the big data, which has attracted the attention and widespread concern of researchers all over the world [1,2,3,4,5,6,7,8,9,10,11,12,13].

Micro-strip filter is a very important component of wireless sensor network (WSN) nodes, which is used to select useful signals and suppress clutter interference signals. Multi-frequency micro-strip filters are required to effectively pick up the signals of each separate frequency band and prevent signal crosstalk between adjacent channels, which requires higher frequency selectivity and out-of-band rejection of the filters [4,5,6]. In recent years, scholars in various countries have carried out in-depth research and proposed various design methods to solve the two core problems in the design of multi-band micro-strip filters [14,15,16,17,18,19,20]: Firstly, multi-point frequency selection can be realized, multi-frequency can work in parallel, and each central frequency point has a certain capacity bandwidth; secondly, the signals in adjacent frequency bands must be effectively isolated and can not interfere with each other, and the out-of-band rejection performance of the filter is reliable. There are two main methods for designing multi-band filters [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35]: Firstly, based on the combination of multi-band filters, including the cascade of broadband filters and notch filters, and the parallel connection of multiple filters in different frequency bands; secondly, based on the parasitic frequency of resonators, a multi-band filter is designed.

Traditional Butterworth, Chebyshev, and elliptic function filters can increase the design order of the filter to improve the filter’s out-of-band suppression and frequency selectivity, but the designed filter is usually single-band, not only the size is large, but also the filter’s in-band characteristics are not ideal [36]. Generalized Chebyshev filter (GCF) is also known as quasi-elliptic function filter. It is between Chebyshev and elliptic function filter, which has excellent in-band characteristics and steep edge characteristics [37, 38]. The transmission zeros of generalized Chebyshev filter can be flexibly controlled, which can be used to improve filter selectivity and stop-band isolation. In order to improve the in-band and out-of-band performance of the filter, the communication band of the filter is extended from a single band to a dual band. Based on the synthesis theory of generalized Chebyshev filter, the frequency transformation is carried out with low-frequency prototype, and the design from low-frequency prototype to dual-band filter is realized with a cross-coupling synthesis theory.

2 Methodology

2.1 Frequency conversion method of symmetrical dual band

The single-band filter model is transformed into a dual-band filter and has to undergo two frequency conversion. Firstly, the normalized single low-pass filter is transformed into a normalized dual-band filter, and then, the normalized dual-band filter is transformed into a dual-band filter of actual frequency by one transformation. Figure 1 shows that three frequency variables are applied in the frequency conversion process, one is the normalized low-pass frequency variable Ω, the other is the dual-band normalized intermediate variable Ω′, and the third is the actual frequency variable ω.

Fig. 1
figure 1

Schematic diagram of dual-passband frequency conversionb. a The Ω domain. b The Ω′ domain. c The ω domain

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The transmission function of the generalized Chebyshev N-order low-pass filter [35] is shown as below

$$ {T}^2(s)={S}_{21}^2(s)=\frac{1}{1+{\varepsilon}^2{C}_N^2(s)} $$
(1)

Among them, s = jΩ, ε is the ripple coefficients in the passband and CN(s) is the characteristic function of the generalized Chebyshev low-pass filter.

The transmission zeros and reflection zeros of the generalized Chebyshev low-pass filter are described.

$$ {C}_N\left(\omega \right)=\frac{F\left(\omega \right)}{P\left(\omega \right)}=\frac{\prod_{i=1}^N\left(s-{s}_{pi}\right)}{\prod_{j=1}^K\left(s-{s}_{zj}\right)}\left(K\le N\right) $$
(2)

Among them, spi is the transmission pole, also known as the reflection zero, that is the filter power optimal transmission point and szj is the transmission zero.

It is assumed that the transmission function of the generalized Chebyshev low-pass filter is about Ω axis symmetry. The dual-band frequency conversion process is shown in Fig. 1. Thus, the number of transmission zeros and reflection zeros of the generalized Chebyshev dual-band filter obtained by frequency transformation is 2K and 2N, respectively. The low passband frequency range js' is −j to\( -j{\Omega}_k^{\prime } \), and the high passband frequency range js' is \( j{\Omega}_k^{\prime } \) to j.

Firstly, the normalized frequency conversion from single passband to double passband is realized. The frequency conversion equations such as Eqs. (3) and (4) are given.

$$ s=\frac{s^{\prime }}{a_1}+\frac{a_2}{s^{\prime }}\kern6.12em \left({\Omega}^{\prime }>0\right) $$
(3)
$$ s=\hbox{-} \left(\frac{s^{\prime }}{a_1}+\frac{a_2}{s^{\prime }}\right)\kern4.5em \left({\Omega}^{\prime }<0\right) $$
(4)

s′ is a frequency variable that is mapped from the prototype s plane. As shown in Fig. 1, when s changes from normalized frequency − 1 to 1 in the Ω domain, it maps to s′ from normalized frequency \( {\Omega}_k^{\prime } \) to 1 in the Ω′ > 0 domain; when s changes from normalized frequency − 1 to 1 in the Ω domain, it maps to s′ from normalized frequency \( {\Omega}_k^{\prime } \) to 1 in the Ω′ > 0 domain. Degenerate (3) into

$$ {s^{\prime}}^2-{sa}_1{s}^{\prime }+{a}_1{a}_2=0 $$
(5)

The following Eq. (6) can be obtained by expression (5) of which s = jΩ.

$$ {s}^{\prime }=\frac{j\Omega {a}_1\pm {ja}_1\sqrt{{\left(\Omega \right)}^2+\frac{4{a}_2}{a_1}}}{2} $$
(6)

When s = j,  s′ = j or when \( s=-j,\kern0.5em {s}^{\prime }=j{\Omega}_k^{\prime } \), the Eq. (7) can be obtained from the Eq. (6).

$$ {\displaystyle \begin{array}{l}1=\frac{a_1}{2}\left(1+\sqrt{1+4{a}_2/{a}_1}\right)\\ {}{\Omega_k}^{\prime }=\frac{a_1}{2}\left(1-\sqrt{1+4{a}_2/{a}_1}\right)\end{array}} $$
(7)

We can get s = j when s′ = j and \( {s}^{\prime }=-j{\Omega}_k^{\prime },\kern0.5em s=-j \) when s′ =  − j and \( {s}^{\prime }=j{\Omega}_k^{\prime } \) from Fig. 1. Therefore, the following expressions can be obtained.

$$ {a}_1=1\hbox{-} {\Omega_k}^{\prime}\kern1em ,\kern1em {a}_2=\frac{{\Omega_k}^{\prime }}{1\hbox{-} {\Omega_k}^{\prime }} $$
(8)

Thus, the normalized single passband to normalized dual-band frequency conversion can be easily realized by expressions (7) and (8).

Secondly, the normalized dual-band to the actual frequency of the dual-band transformation, that is, the frequency transform domain from Ω′ domain to ω domain transformation. The transformation equation is

$$ {s}^{\prime }=\frac{\varpi }{l_1}+\frac{l_2}{\varpi}\left(\omega >0\right) $$
(9)

Among it, ϖ = jω. We can see that values of 1, \( {\Omega}_k^{\prime },\kern0.5em -{\Omega}_k^{\prime } \), and − 1 in the Ω′ domain are mapped, respectively, to ωk4,  ωk3,  ωk2, and ωk1 of the ω domain from Fig. 1.

We can get the expressions of l1,  l2, and \( {\Omega}_k^{\prime } \).

$$ {l}_1={\omega}_{k4}-{\omega}_{k1},\kern1em {l}_2=\frac{\omega_{k4}{\omega}_{k1}}{\omega_{k4}-{\omega}_{k1}},\kern1em {\Omega_k}^{\prime }=\frac{\omega_{k3}-{\omega}_{k2}}{\omega_{k4}-{\omega}_{k1}} $$
(10)

2.2 Synthesis method of N-order cross-coupling matrix

With the development of communication technology, the spectrum is becoming more and more crowded, and the technical specifications of the filter, especially the rectangular requirements, are becoming more and more stringent. Traditional Butterworth and Chebyshev filters have been unable to meet the requirements. In order to improve the selectivity and out-of-band isolation of filters, transmission zeros are usually introduced into filters, which are generated by cross-coupling between non-adjacent resonators [30,31,32,33,34,35]. The cross-coupling filter with finite transmission zeros is the most common choice. Generalized Chebyshev function is usually used to implement it.

The lumped parameter equivalent circuit and equivalent network parameters of the coupling filter are shown in Fig. 2a and b, respectively. According to the Kirchhoff theorem that the sum of the voltage is zero along the loop, the voltage of each loop is described Eq. (11).

$$ \left\{\begin{array}{l}\left({R}_1+ j\omega {L}_1+1/ j\omega {C}_1\right)\cdot {i}_1-j{\mathrm{L}}_{12}\cdot {i}_2\cdots -{jL}_{1n}\cdot {i}_n={e}_s\\ {}-{jL}_{12}\cdot {i}_1+\left( j\omega {L}_2+1/ j\omega {C}_2\right)\cdot {i}_2-{jL}_{2n}\cdot {i}_n=0\\ {}\kern1.25em \vdots \\ {}-{jL}_{n1}\cdot {i}_1-{jL}_{n2}\cdot {i}_2\cdots +\left({R}_2+ j\omega {L}_n+1/ j\omega {C}_n\right)\cdot {i}_n=0\end{array}\right. $$
(11)
Fig. 2
figure 2

Equivalent circuit (a) and equivalent network parameters (b) of the loop equation of lumped parameter of N-order coupling filter

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The specific method of treatment is shown in document [15], Eq. (11) which is expressed by matrix as Eq. (12).

$$ \left[Z\right]\cdot \left[I\right]=\left[e\right] $$
(12)

[Z] is N × N impedance matrix. Each resonator of synchronous tuning filter has the same resonance frequency \( {\omega}_0=1/\sqrt{LC} \), and among it, L = L1 = L2 = ⋯Ln,  C = C1 = C2 = ⋯Cn. FBW = Δω/ω0 is the relative bandwidth. The normalized impedance matrix is \( \left[\overline{Z}\right]=\left[Z\right]/\left({\omega}_0L\cdot FBW\right) \).

$$ \left[\overline{Z}\right]=\left[\begin{array}{cccc}\frac{R_1}{\omega_0L\cdot FBW}+p& -j\frac{\omega {L}_{12}}{\omega_0L\cdot FBW}& \cdots & -j\frac{\omega {L}_{1n}}{\omega_0L\cdot FBW}\\ {}-j\frac{\omega {L}_{21}}{\omega_0L\cdot FBW}& p& \cdots & \vdots \\ {}\vdots & \vdots & \vdots & \vdots \\ {}-j\frac{\omega {L}_{n1}}{\omega_0L\cdot FBW}& -j\frac{\omega {L}_{n2}}{\omega_0L\cdot FBW}& \cdots & \frac{R_n}{\omega_0L\cdot FBW}+p\end{array}\right]\kern3em p=j\frac{1}{FBW}\left(\frac{\omega }{\omega_0}-\frac{\omega_0}{\omega}\right) $$
(13)

The external quality factor is defined as Qei = Ri/ω0L(i = 1, N) and the coupling coefficient isMij = Lij/L. While ω0/ω ≈ 1 in the narrowband filter, there are

$$ \left[\overline{Z}\right]=\left[\begin{array}{cccc}\frac{1}{q_{ei}}+p& -{jm}_{12}& \cdots & -{jm}_{1n}\\ {}-{jm}_{21}& p& \cdots & \vdots \\ {}\vdots & \vdots & \vdots & \vdots \\ {}-{jm}_{n1}& -{jm}_{n2}& \cdots & \frac{1}{q_{en}}+p\end{array}\right] $$
(14)

Among it, qei = Qei • FBW(i = 1.2),  mij = Mij/FBW, to I1 = i1,  In =  − in in Fig. 2b, then we can get the Eq. (15).

$$ {\displaystyle \begin{array}{l}{a}_1=\frac{e_s}{2\sqrt{R_1}}\kern5.5em {b}_1=\frac{e_s-2{i}_1{R}_1}{2\sqrt{R_1}}\\ {}{a}_2=0\kern7.5em {b}_2={i}_n\sqrt{R_n}\end{array}} $$
(15)

After synthesizing the coupling matrix, the coupling matrix is deformed to correspond to the actual circuit model structure. This is advantageous to the actual physical circuit design, in which the negative coupling coefficient in the coupling matrix represents the capacitance coupling in the cross-coupling of the adjacent resonator circuit, and the positive coupling coefficient in the coupling matrix represents the inductive coupling in the cross-coupling of the adjacent resonator circuit.

3 Results and discussion

3.1 Basic parameters of dual-band filter

Based on the previous generalized Chebyshev filter function synthesis theory of dual-band filter, a symmetrical dual-band filter (SDF) are designed. Assuming that the frequency range of the low passband is 2.9–2.95 GHz, the frequency range of the high passband is 3.05–3.10 GHz and the in-band return loss is RL = 22 dB.

The dual-band is symmetrical with respect to the center of ω0 = 3 GHz, and the four transmission zeros of the single-pass low-pass filter are 1.5j,   − 1.5j,  j∞, and −j∞.

ωk1 = 2.90,  ωk2 = 2.95,  ωk3 = 3.05, and ωk4 = 3.10 can be calculated from the conditions given above. According to Eq. (10), we can obtain \( {\Omega}_k^{\prime }=0.5 \). According to Eq. (8), we can obtain a1 = 0.5 and a2 = 1.

3.2 Design and simulation of symmetrical dual-band filter

Based on the generalized Chebyshev filter synthesis method described in the previous section, we can obtain the roots of these functions P(Ω),  F(Ω), and E(Ω), as shown in Table 1.

Table 1 Polynomial roots of the four-order single-passband filter
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$$ P\left(\Omega \right)={\Omega}^2-2.25 $$
$$ F\left(\Omega \right)={\Omega}^4-1.730{\Omega}^2+0.1641\kern0.5em \varepsilon -0.8666 $$
$$ E\left(\Omega \right)={\Omega}^4-j2.423{\Omega}^3-4.009{\Omega}^2+j3.81\Omega +2.601 $$

We can plot the response curve of the S parameter, as shown in Fig. 3. The coupling matrix [16] deduced from short-circuit admittance parameters is:

$$ m=\left[\begin{array}{cccc}0& 0.861& 0& -0.476\\ {}0.861& 0& -0.911& 0\\ {}0& 0.911& 0& -0.861\\ {}-0.476& 0& -0.861& 0\end{array}\right] $$
Fig. 3
figure 3

S parameters curve of four-order single-passband filter

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Now, by using the frequency mapping equation, the reflection and transmission zeros of single-passband normalized generalized Chebyshev function filters and a1 = 0.5,  a2 = 1 are substituted in Eq. (6), we can obtain:

$$ {s}^{\prime }=\frac{j\Omega {a}_1\pm {ja}_1\sqrt{{\left(\Omega \right)}^2+\frac{4{a}_2}{a_1}}}{2}=\frac{j0.5\left(\Omega \pm \sqrt{{\left(\Omega \right)}^2+8}\right)}{2} $$
(16)

The transmission and reflection zeros of dual-band filters are obtained by using Eq. (16). Then, the root of E(s′) is obtained by using the generalized Chebyshev filter function synthesis of a symmetrical dual-band. The roots of P(Ω′),  F(Ω′), and E(Ω′) are listed in Table 2.

$$ P\left({\Omega}^{\prime}\right)={\Omega^{\prime}}^6+1.5625{\Omega^{\prime}}^4+0.25{\Omega^{\prime}}^2 $$
$$ F\left({\Omega}^{\prime}\right)={\Omega^{\prime}}^8+2.2682{\Omega^{\prime}}^6+1.7785{\Omega^{\prime}}^4+0.56706{\Omega^{\prime}}^2+0.0625 $$
$$ {\displaystyle \begin{array}{l}E\left({\Omega}^{\hbox{'}}\right)={\Omega^{\hbox{'}}}^8+1.119{\Omega^{\hbox{'}}}^7+2.895{\Omega^{\hbox{'}}}^6+2.087{\Omega^{\hbox{'}}}^5+2.524{\Omega^{\hbox{'}}}^4+1.044{\Omega^{\hbox{'}}}^3\\ {}\kern.6em +0.723{\Omega^{\hbox{'}}}^2+0.1399{\Omega^{\hbox{'}}}^1+0.062\end{array}} $$
Table 2 Polynomial roots of the eight-order dual-band filter
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Especially, the filter is a four-order filter, with two finite transmission zeros, and two transmission zeros located at the positive and negative infinity. In the case of dual-passband transformation, the transmission zeros at infinity of single passband correspond to the transmission zeros at zero of dual passband. After the transmission and reflection zeros of the two passbands are obtained, the expressions of each polynomial of the two passbands can be obtained by using the synthesis method of the generalized Chebyshev filter function.

Thus, the normalized response curve of the dual band-pass filter can be obtained, as shown in Fig. 4.

Fig. 4
figure 4

Normalized S parameters response curve of dual-passband filter

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Finally, Eq. (10) is used to calculate l1 and l2, and the Eq. (17) are used to map the frequency domain from Ω′ to ω domain. Then, the S parameter curve in the actual frequency domain is obtained as shown in Fig. 5.

$$ \omega =\frac{l_1}{2}\left({\Omega}^{\prime }+\sqrt{1+4{l}_2/{l}_1}\right) $$
(17)
Fig. 5
figure 5

S parameters response curve of dual-passband actual frequency

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The coupling matrix obtained from the coefficients of P(Ω′),  F(Ω′), and E(Ω′) are then rotated to eliminate the element. The folded N + 2-order, that is, the tenth-order coupling matrix is.

$$ m=\left[\begin{array}{l}\begin{array}{cccc}0& 0.7481& 0.0000& 0.0000\\ {}0.7481& 0.0000& 0.8215& -0.0000\\ {}0.0000& 0.8215& -0.0000& 0.4075\\ {}0.0000& 0.0000& 0.4075& 0\end{array}\kern2.12em \begin{array}{cccc}0.0000& 0.0000& 0.0000& 0.0000\\ {}0.0000& 0.0000& -0.0000& -0.0000\\ {}0& 0& 0& 0.1129\\ {}0.4750& 0& 0.5746& 0.0000\end{array}\kern4.5em \begin{array}{c}0.0000\\ {}-0.2045\\ {}0.0000\\ {}-0.0000\end{array}\kern2.12em \begin{array}{c}0\\ {}0.0000\\ {}0.0000\\ {}0.0000\end{array}\\ {}\begin{array}{cccc}0.0000& 0.0000& -0.0000& 0.4750\\ {}0.0000& -0.0000& 0.0000& 0.0000\\ {}0.0000& 0.0000& -0.0000& 0.5746\\ {}0.0000& 0.0000& 0.1129& 0.0000\end{array}\kern2em \begin{array}{cccc}0.0000& -0.2519& 0.0000& -0.0000\\ {}-0.2519& 0.0000& 0.4750& 0.0000\\ {}-0.0000& 0.4750& -0.0000& 0.4075\\ {}-0.0000& 0.0000& 0.4075& -0.0000\end{array}\kern2.5em \begin{array}{c}0.0000\\ {}0.0000\\ {}-0.0000\\ {}0.8215\end{array}\kern2em \begin{array}{c}0.0000\\ {}-0.0000\\ {}0.0000\\ {}-0.0000\end{array}\\ {}\begin{array}{cc}0.0000& -0.2045\\ {}0& 0.0000\end{array}\kern2em \begin{array}{cc}-0.0000& -0.0000\\ {}0.0000& 0.0000\end{array}\kern1.5em \begin{array}{cc}0.0000& 0.0000\\ {}0.0000& -0.0000\end{array}\kern1.86em \begin{array}{cc}0.0000& 0.8215\\ {}0.0000& -0.0000\end{array}\kern3.48em \begin{array}{cc}0.0000& 0.7481\\ {}0.7481& 0\end{array}\end{array}\right] $$

4 Conclusion

This paper presented an experimental study on the micro-strip dual-band filter based on wireless sensor network nodes. Firstly, according to the order, the position of the transmission zeros, and the ripple coefficients in the band of the filter to be designed, the generalized Chebyshev filter function synthesis method is applied to synthesize a single-passband filter by combining the relationship between the short-circuit admittance parameters. The coupling matrix of the single-passband filter is obtained. Secondly, the generalized Chebyshev function polynomial is constructed according to the symmetrical frequency conversion equations from single-band to dual-band, and the coupling matrix of the dual-band filter is synthesized by the relationship between the generalized Chebyshev function polynomial and the short-circuit admittance parameters. Finally, the normalized frequency domain is mapped to the actual frequency domain, and the S parameter response curve in the actual frequency domain is obtained. By flexibly controlling the transmission zeros of the generalized Chebyshev filter, the selectivity and stop-band isolation of the filter can be effectively improved, and this method can design multi-band filters with excellent performance and has a positive guiding role in the physical circuit design of micro-strip filters based on wireless sensor network nodes.