# A Novel Active Polyphase Filter Employing Frequency-Dependent Image Rejection Enhancement Technique
## 3 Circuit Implementation  
In APPFs, both \(\omega_{H}\) and \(\omega_{L}\) are determined by the transistor transconductance and the main capacitance. In contrast to the capacitance value, which can only be tuned discretely by a switched-capacitor bank, the transconductance value can be easily tuned continuously by changing the current flowing through the transistor. Therefore, it is a more reliable circuit implementation scheme to adjust the transistor transconductance value to achieve the proposed image rejection enhancement technique. Figure 8: Processing mechanism to phase imbalance of the input signal: (**a**) the quadrature input signal is less than \(90^{\circ}\); and (**b**) the quadrature input signal is greater than \(90^{\circ}\). Figure 7: Compensation mechanism of the proposed enhancement technique. Figure 9 shows the novel single-stage LHF topology, consisting of low-pass modules and high-pass modules. The low-pass module consists of transistors M\({}_{1}\), M\({}_{2}\), M\({}_{3}\), and the main capacitor \(C_{L}\). The output current \(i_{L}\) of the low-pass module is converted to voltage by connecting it to a diode-connected transistor M\({}_{Load}\). The transfer function of the low-pass module \(H_{L}(s)\) can be written as  
\[H_{L}(s)=\frac{g_{m1}}{g_{mLoad}}\cdot\frac{g_{m2}/C_{L}}{s+g_{m2}/C_{L}} \tag{12}\]  
where \(g_{m1}\), \(g_{m2}\), and \(g_{mLoad}\) are the transconductance values of M\({}_{1}\), M\({}_{2}\), and M\({}_{Load}\), respectively. The high-pass module consists of transistors M\({}_{1}\), M\({}_{2}\), M\({}_{3}\), M\({}_{4}\), M\({}_{5}\), and the main capacitor \(C_{L}\). After taking into account the parasitic capacitance \(C_{P}\) at the node N, the transfer function of the high-pass module \(H_{H}(s)\) can be derived as  
\[ \begin{array}{ll}H_{H}(s)&=\frac{g_{m1}}{g_{mLoad}}\cdot\frac{g_{m2}/C_{P}}{ s+g_{m2}/C_{P}}-\frac{g_{m1}}{g_{mLoad}}\cdot\frac{g_{m1}/C_{L}}{s+g_{m2}/C_{L}}\\ &=\frac{g_{m1}}{g_{mLoad}}\cdot\frac{s}{s+g_{m4}/C_{L}}\cdot\frac{g_{m2}/C_{P} -g_{m4}/C_{L}}{s+g_{m2}/C_{P}}\end{array}  \tag{13}\]  
where \(g_{m4}\) is the transconductance value of M\({}_{4}\). In the novel single-stage LHF topology, \(\omega_{L}\), \(\omega_{H}\), and \(\omega_{P}\) are generated by \(g_{m2}\)/\(C_{L}\), \(g_{m4}\)/\(C_{L}\), and \(g_{m2}\)/\(C_{P}\), respectively. In order to implement the proposed enhancement technique, the transistors M\({}_{5}\) are added to the high-pass modules to introduce the additional compensation currents to change the magnitude of the currents flowing through M\({}_{4}\). In this way, the required \(\omega_{H}\) can be obtained while \(\omega_{L}\) is not affected in any way. The transistors M\({}_{3}\) are added to make the circuit more symmetrical and matched and to help compensate for the phase imbalance \(\theta_{IF}\). In order to satisfy the condition between \(\omega_{H}\) and \(\omega_{L}\) in (11), \(g_{m4}\) and \(g_{m2}\) in the proposed APPF must satisfy the proportional relationship determined by \(\omega_{P}\) and the operating frequency \(\omega_{IF}\), as shown in Equation (14). \[\frac{g_{m4}}{g_{m2}}=\frac{\omega_{P}+\omega_{IF}}{\omega_{P}-\omega_{IF}} \tag{14}\]  
The relationship between the transconductance value \(g_{m}\) of a transistor operating in the saturated region and the current \(I\) flowing through the transistor is as follows:  
\[g_{m}=\sqrt{2\mu_{n}C_{ox}\frac{W}{L}I} \tag{15}\]  
where \(\mu_{n}\), \(C_{ox}\), and \(W/L\) are the mobility of the electrons, the gate oxide capacitance per unit area, and the aspect ratio of the transistor, respectively. It can be seen that by controlling transistor M\({}_{5}\) to vary the additional compensation current \(I_{COM}\), the current \(I_{4}\) flowing through transistor M\({}_{4}\) can be varied to adjust \(g_{m4}\). Transistors M\({}_{2}\) and M\({}_{4}\) have the same physical parameters such as the aspect ratio.