**3. Synthesis of the constant-BW filter using EVCM**

With the introduction of the new matrix, the tunable filter can correspond to an EVCM uniquely. Thus, the tunable filter with constant BW can be synthesized with a few steps of the mathematical manipulations. First, for a tunable resonator, the adjustment of the *CF* is realized by tuning the loading effect, which can be treated as adding or subtracting the susceptance from the resonator. Thus, the tunable parameter of a resonator can be modeled by the corresponding diagonal element of an EVCM as:

$$m\_{tt} = \left(\frac{f\_d}{f\_t} - \frac{f\_t}{f\_d}\right) \tag{11}$$

For the tunable coupled-resonator filters, it is reasonable to define *mtt*<sup>1</sup> = *mtt*<sup>2</sup> = *mtt*<sup>3</sup> = … = *mtt*, and thus, when the filter is adjusted, the frequencies of all the resonators will be tuned with the same tuning step. Additionally, to maintain the constant BW over a wide tuning range, when adjusting CF, the entries of the EVCM

### *Tunable Filter DOI: http://dx.doi.org/10.5772/intechopen.104391*

(including the external quality factor *Qe*) are required to change in a specific manner synchronously.

Based on the earlier discussion, one may assume that the EVCMs are the linear functions of the variable *mtt*, and thus the EVCM [*m*Δ]*<sup>t</sup>* is converted as (12, 13). [*m*Δ] is the frequency-fixed part of the EVCM, which is unchanged when the frequency is tuned*.* [*c*] is a factor matrix and determines the passband variation rate of the filter. When tuning CF, the total matrix varies as predefined by [*c*], which results in a tunable passband that varies in a linear function according to the predefined response and BW variation:

$$[m\_{\Delta}]\_t = \begin{bmatrix} m\_{11} + c\_{11}m\_{tt} & m\_{12} + c\_{12}m\_{kk} & \cdots & m\_{1t} + c\_{1t}m\_{tt} \\\\ m\_{21} + c\_{21}m\_{tt} & m\_{22} + c\_{22}m\_{tt} & \cdots & m\_{2t} + c\_{2t}m\_{tt} \\\\ \vdots & \vdots & \vdots & \vdots \\\\ m\_{n1} + c\_{n1}m\_{tt} & m\_{n2} + c\_{n2}m\_{tt} & \cdots & m\_{kk} + c\_{kk}(m\_{tt}) \end{bmatrix} = ([m\_{\Delta}] + m\_{tt}[c]),$$
 
$$\mathbf{Q}\_{\varepsilon} = \Delta \mathbf{Q}\_{\varepsilon} + m\_{kk}c\_{Q}$$

$$\begin{aligned} [c] = \begin{bmatrix} c\_{11} & c\_{12} & \cdots & c\_{1n} \\\\ c\_{21} & c\_{22} & \cdots & c\_{2n} \\\\ \vdots & \vdots & \vdots & \vdots \\\\ c\_{n1} & c\_{n2} & \cdots & c\_{kk} \end{bmatrix} \tag{13} \\ \tag{13} \\ \vdots \\ \vdots \\ \vdots \end{aligned} \tag{14}$$

(12)

The linear EVCM that represents the tunable filter with a specified tuning range and reflects the passband variation or the passband tuning behavior is defined. Now, extracting parameters of the EVCM from the general filtering function is of great importance to design the tunable passband with constant *BW.* The extraction method is derived as follows.

First, the tuning range from *fl* to *fh* is given, and then the mapping frequency can be defined as *fd* = (*fl* + *fh*)/2. The RL and BW are, respectively, prescribed at *RL* and *ABW*. Then, the conventional coupling matrix [*M*]*<sup>d</sup>* is extracted at *fd*. Note here that the diagonal elements are all zeros because the resonant frequencies of all resonators are the same (with symmetric distribution) for the simpler case demonstration:

$$\begin{Bmatrix} \begin{Bmatrix} M \end{Bmatrix}\_d = \begin{bmatrix} \mathbf{0} & M\_{12} & M\_{13} & \dots & M\_{1N} \\\\ M\_{21} & \mathbf{0} & M\_{23} & & M\_{2N} \\\\ M\_{31} & M\_{32} & \ddots & & \vdots \\\\ \vdots & & \ddots & M\_{N-1,N} \\\\ M\_{N1} & & & \mathbf{0} \end{bmatrix} \\\\ \mathbf{q}\_{\varepsilon} \end{Bmatrix} \tag{14}$$

The coupling matrix response can be mapped to *fd*, scaled by *ABW*/*fl* and moved to *fl*. Thus, the coupling matrix is expressed as:

$$\begin{cases} [m](m\_{\rm tt} = m\_{\rm tt}) = [\mathcal{M}]\_d \cdot \left(AB\mathcal{W}/f\_l\right) + [\mathcal{W}] \cdot m\_{\rm tt} \\\\ \mathcal{Q}\_{el} = q\_\epsilon / \left(AB\mathcal{W}/f\_l\right) \end{cases} \tag{15}$$

[*W*] is an identity matrix. Similarly, the coupling matrix response can be mapped to *fd*, scaled by *ABW*/*fh*, and moved to *fh*. Then the coupling matrix is written as:

$$\begin{cases} [\boldsymbol{m}](\boldsymbol{m}\_{\rm tt} = \boldsymbol{m}\_{\rm tt}) = [\boldsymbol{M}]\_{d} \cdot \left( \boldsymbol{ABW} / f\_{\boldsymbol{h}} \right) + [\boldsymbol{W}] \cdot \boldsymbol{m}\_{\rm tt} \\\\ \boldsymbol{Q}\_{\rm sh} = \boldsymbol{q}\_{\epsilon} / \left( \boldsymbol{ABW} / f\_{\boldsymbol{h}} \right) \end{cases} \tag{16}$$

Using linear interpolation, the EVCM can be extracted as:

$$\begin{cases} [\boldsymbol{m}\_{\Delta}]\_t = ([\boldsymbol{m}\_{\Delta}] + \boldsymbol{m}\_{\boldsymbol{tt}}[\boldsymbol{c}]) \\ \boldsymbol{Q}\_{\boldsymbol{\epsilon}} = \Delta \boldsymbol{Q}\_{\boldsymbol{\epsilon}} + \boldsymbol{m}\_{\boldsymbol{tt}} \boldsymbol{c}\_{\boldsymbol{Q}} \end{cases} \tag{17}$$

where

**Figure 15.** *Extracted 100% tunable 4th-/6th-/8th-order responses with constant 100-MHz ABW.*

*Tunable Filter DOI: http://dx.doi.org/10.5772/intechopen.104391*

$$\begin{cases} [m\_{\Delta}] = ABW \cdot [M]\_d \cdot \left(\frac{\left(f\_h + f\_l\right)}{\left(f\_d^{\prime} + f\_h f\_l\right)}\right) \\\\ [c] = [M]\_d ABW \cdot \frac{f\_d \left(f\_l; f\_h\right)}{\left(f\_1 f\_d^{\prime} - f\_h f\_d^{\prime} - f\_l f\_h^{\prime} + f\_l^{\prime} f\_h\right)} + [W] \\\\ \Delta Q\_{\epsilon} = \frac{q\_{\epsilon}}{ABW} \left(\frac{f\_d^{\prime 2} \left(f\_h + f\_l\right)}{f\_d^{\prime 2} + f\_h f\_l}\right) \\\\ \varepsilon\_{Q} = \frac{q\_{\epsilon}}{ABW} \left(-\frac{\left(f\_d f\_h f\_l\right)}{f\_d^{\prime 2} + f\_h f\_l}\right) \end{cases} \tag{18}$$

Now the extraction process is completed and the constant-BW tunable filter is synthesized by the EVCM.

For illustration, the 4th-/6th-/8th-order fully canonical folded filters are extracted with 20-dB RL, a 100% frequency tuning range (from 0.5 GHz to 1.5 GHz), and a constant 100 MHz *ABW*. The extracted EVCMs are given as follows:

Four-pole filter:

$$\begin{cases} [m]\_{4-pole} = \begin{cases} \begin{array}{c} m\_{12} = 0.09259 + m\_{kk} \cdot 0.0463\\ m\_{23} = 0.095338 + m\_{kk} \cdot 0.04767\\ m\_{14} = -0.04064 - m\_{kk} \cdot 0.0203 \end{array} \end{cases} \tag{19}$$

$$Q\_{\ast 4-pole} = \mathbf{11.114465} - m\_{kk} \cdot \mathbf{4.1679}, m\_{\ast} = -5.15 \mathbf{e} \cdot \mathbf{04}$$

Six-pole filter:

$$\begin{cases} \begin{aligned} \label{eq:10} \begin{cases} \begin{aligned} m\_{12} &= 0.09487 + m\_{kk} \cdot 0.04744 \\ m\_{23} &= 0.066177 + m\_{kk} \cdot 0.03309 \\\\ m\_{34} &= 0.080465 + m\_{kk} \cdot 0.04023 \\\\ m\_{25} &= -0.01828 - m\_{kk} \cdot 0.0091 \\\\ m\_{16} &= 0.00137 + m\_{kk} \cdot (6.867 \text{e-} 04) \end{aligned} \end{cases} \end{cases} \end{cases} \end{cases} \tag{20}$$

Eight-pole filter:

½ � *<sup>m</sup>* <sup>8</sup>�*pole* <sup>¼</sup> *m*<sup>12</sup> ¼ 0*:*09232 þ *mkk* � 0*:*04616 *m*<sup>23</sup> ¼ 0*:*0657 þ *mkk* � 0*:*03285 *m*<sup>34</sup> ¼ 0*:*0469 þ *mkk* � 0*:*02345 *m*<sup>45</sup> ¼ 0*:*09855 þ *mkk* � 0*:*04927 *m*<sup>36</sup> ¼ �0*:*048 � *mkk* � 0*:*02404 *m*<sup>27</sup> ¼ 0*:*009 þ *mkk* � 0*:*0045 *<sup>m</sup>*<sup>18</sup> ¼ �7*:*28e‐<sup>04</sup> � *mkk* � <sup>3</sup>*:*64e � <sup>04</sup> 8 >>>>>>>>>>>>>>< >>>>>>>>>>>>>>: *Qe*<sup>8</sup>�*pole* ¼ 11*:*84219 � *mkk* � 4*:*4408, *msl* ¼ 2*:*057e � 05 8 >>>>>>>>>>>>>>>>>< >>>>>>>>>>>>>>>>>: (21)

**161**

**Figure 15** presents the calculation responses of three extracted EVCMs. It can be seen that the prescribed responses are tuned from 0.5GHz to 1.5 GHz, and their responses are kept nearly constant. Even though such a wide tuning range and 8thorder function are predefined, the maximum estimation error of this synthesis method only affects the RL. When the frequency of the passband is tuned closer to the middle point of the tuning range [*fd* = (*fh*+*fl*)/2], the most significant RL deviation will be introduced, which will lead to deterioration of the high-order filter. However, it is still acceptable for the high-order tunable design with a wide FTR (100%).
