# A Novel Active Polyphase Filter Employing Frequency-Dependent Image Rejection Enhancement Technique
## 2 Circuit Analysis
### Theoretical Model of APPF
\begin{array}{ll}IRR&=\frac{\left|\hat{s}_{signal_{image}} \right|}{\left|\hat{s}_{signal_{doind}}\right|}=\frac{\sqrt{A^{2}+\left[A(1+ \Delta A)\right]^{2}+2A^{2}(1+\Delta A)\cos(\pi-\theta)}}{\sqrt{A^{2}+\left[A (1+\Delta A)\right]^{2}+2A^{2}(1+\Delta A)\cos\theta}}\\ \\ &=\left[\frac{(1+\Delta A)^{2}-2(1+\Delta A)\cos\theta+1}{(1+\Delta A)^{2}+2( 1+\Delta A)\cos\theta+1}\right]^{\frac{1}{2}}\\ \end{array}  \tag{6}\]  
Figure 1: Theoretical model of APPF. Figure 2: Block diagram of a typical low-IF receiver. Figure 4 shows the results of the IRR under various amplitude and phase imbalances, and the numbers on the curves denote the IRR values in decibels. A slight imbalance in amplitude or phase will cause a serious degradation of the IRR. Figure 5 shows the topology of the low-and-high-pass filter (LHF) unit, which consists of low-pass and high-pass transconductance modules in a conventional APPF [24]. Compared to the ideal LPF and HPF transfer functions, the LHF transfer function implemented in the actual circuit deviates from the ideal mathematical model due to the parasitic effect of the physical components. The secondary pole generated by the parasitic capacitance \(C_{P}\) at node N is the most serious inherent non-ideal factor in the APPF. The mismatches of gains and pole frequencies between the LPFs and HPFs caused by \(C_{P}\) will directly result in amplitude and phase imbalances. After considering the secondary pole, the transfer function \(H(s)\) of the APPF in Equation (2) can be rewritten as  
\[ \begin{array}{ll}H(s)&=A_{L}\frac{\omega_{L}}{s+\omega_{L}}+j\Big{(}A_{H} \frac{\omega_{P}}{s+\omega_{P}}-A_{H}\frac{\omega_{H}}{s+\omega_{H}}\Big{)}\\ &=A_{L}\frac{\omega_{L}}{s+\omega_{L}}+j\Big{(}A_{H}\frac{C_{L}-C_{P}}{C_{L}} \Big{)}\Big{(}\frac{s}{s+\omega_{P}}\cdot\frac{\omega_{P}}{s+\omega_{P}}\Big{)} \end{array}  \tag{7}\]  
where \(\omega_{P}\) and \(C_{L}\) are the secondary pole frequency and the main capacitance, respectively. In Figure 5, the formula for the dominant pole frequency and the secondary pole frequency are \(\omega_{L}=\omega_{H}=g_{m2}/C_{L}\) and \(\omega_{P}=g_{m2}/C_{P}\), respectively, where \(g_{m2}\) is the transconductance value of transistor M\({}_{2}\). The gain of the HPF is no longer \(A_{H}\), but becomes the product of \(A_{H}\) and the coefficient \((C_{L}-C_{P})/C_{L}\). Since the coefficient \((C_{L}-C_{P})/C_{L}\) is constantly less than one, the gain of the LPF \(A_{L}\) is no longer equal to the gain of the HPF \(A_{H}(C_{L}-C_{P})/C_{L}\). It can be seen from (7) that \(C_{P}\) and \(\omega_{P}\) cause the gain and pole frequency mismatches of the APPF, which will introduce amplitude and phase imbalances. \(C_{P}\) at node N in Figure 5 is composed of the gate-drain parasitic capacitance \(C_{gd}\), the gate-source parasitic capacitance \(C_{gs}\), the drain junction capacitance \(C_{j\mu}\), and the source junction capacitance \(C_{j\beta}\) of the MOSFETs. Furthermore, the layout and routing will also introduce some of the parasitic capacitance at node N. Taking the 180 nm CMOS process as an example, on the premise that the overall power consumption is acceptable, \(g_{m2}\) is set at 500 \(\mu\)S. After using the relatively short transistors to reduce the parasitic capacitance, the value of \(C_{P}\) is still in the tens of femto-farads (ffs). Reducing the channel length of the transistors further not only does not effectively reduce the value of \(C_{P}\) but also the output resistance of the transistors will be smaller, which can produce parasitic zeros that have a negative effect on image rejection. By setting \(C_{P}\) to 20 fF, \(\omega_{P}\) can be obtained in the vicinity of 4 GHz, which is close to the limit of the 180 nm CMOS process. Figure 6 shows the trend of IRR degradation with increasing operating frequency. At relatively high operating frequencies, the ratio of \(C_{P}\) to \(C_{L}\) increases, and the frequencies of \(\omega_{L}\) and \(\omega_{H}\) are closer to the frequency of \(\omega_{P}\).