# A Novel Active Polyphase Filter Employing Frequency-Dependent Image Rejection Enhancement Technique
## 2 Circuit Analysis  
### Theoretical Model of APPF  
Figure 1 shows the theoretical model of the APPF. After the quadrature down-conversion, the desired signal and the image interference are symmetric about the origin in the complex frequency domain and have a completely opposite phase order. In essence, the APPF is a complex notch filter. According to its theoretical model, the transfer function _H(s)_ can be written as  
\[ \begin{array}{ll}H(s)&=\frac{I_{\text{D}}(s)+j\text{O}_{\text{D}}(s)}{I_{ \text{I}}(s)+j\text{O}_{\text{I}}(s)}=\frac{[I_{\text{I}}(s)H_{\text{I}}(s)-Q _{\text{I}}(s)H_{\text{H}}(s)]+j[I_{\text{I}}(s)H_{\text{I}}(s)+Q_{\text{I}}( s)H_{\text{I}}(s)]}{I_{\text{I}}(s)+j\text{O}_{\text{I}}(s)}\\ &=H_{\text{L}}(s)+jH_{\text{H}}(s)\end{array}  \tag{1}\]  
where \(H_{\text{L}}(s)\) and \(H_{\text{H}}(s)\) are the transfer functions of the LPF and high-pass filter (HPF), respectively. The mixed signals consisting of the desired signal and the image interference pass through the LPFs and HPFs in the I/Q channels. The transfer function of the APPF consisting of first-order LPFs and first-order HPFs can be rewritten as  
\[H(s)=A_{L}\frac{\omega_{\text{L}}}{s+\omega_{\text{L}}}+jA_{H}\frac{s}{s+ \omega_{H}} \tag{2}\]  
where \(A_{L}\), \(A_{H}\), \(\omega_{\text{L}}\), and \(\omega_{H}\) are the gains and the pole frequencies of the LPF and HPF, respectively. At the pole frequencies of the LPF and HPF, the phase of the signal lags and leads by 45\({}^{\circ}\), respectively. After combining the output signals of the LPFs and HPFs in the I/Q channels as shown in Figure 1, the image interference in the mixed signal can theoretically be completely rejected, and the desired signal is amplified \(\sqrt{2}\) times. The corresponding mathematical proof can be written as follows:  
\[H(s=j\omega_{\text{IF}})=A\frac{\omega_{\text{IF}}}{j\omega_{\text{IF}}+ \omega_{\text{IF}}}+jA\frac{j\omega_{\text{IF}}}{j\omega_{\text{IF}}+\omega_{ \text{IF}}}=A\frac{\omega_{\text{IF}}-\omega_{\text{IF}}}{j\omega_{\text{IF}}+ \omega_{\text{IF}}}=0 \tag{3}\]  
\[H(s=-j\omega_{\text{IF}})=A\frac{\omega_{\text{IF}}}{-j\omega_{\text{IF}}+ \omega_{\text{IF}}}+jA\frac{-j\omega_{\text{IF}}}{-j\omega_{\text{IF}}+\omega_ {\text{IF}}}=A\frac{\omega_{\text{IF}}+\omega_{\text{IF}}}{-j\omega_{\text{IF} }+\omega_{\text{IF}}}=A\frac{2}{1-j} \tag{4}\]  
\(\omega_{\text{IF}}\) represents the frequency of the image interference in the complex frequency domain, and \(-\omega_{\text{IF}}\) represents the frequency of the desired signal. In addition, the gains and pole frequencies of the LPF and HPF must satisfy the operating conditions of \(A=A_{L}=A_{H}\) and \(\omega_{IF}=\omega_{L}=\omega_{H}\). Under such ideal operating conditions, the IRR, defined as the ratio of the signal strength of the image interference to the signal strength of the desired signal, is 0, as shown in Equation (5). \[IRR(\omega_{IF})=\frac{\left|H(s=j\omega_{IF})\right|}{\left|H(s=-j\omega_{IF}) \right|}=0 \tag{5}\]  
Figure 2 shows the block diagram of a typical single-quadrature low-IF receiver that uses an APPF in the IF processing module to reject image interference and slightly amplify the desired signal. Ideally, the image interference can be rejected to null. However, in the actual circuit, there are non-ideal factors such as the parasitic capacitance in the APPF and the less-than-perfect input signal to the APPF, which lead to the degradation of the image rejection effect. Figure 3 illustrates the principle of image rejection degradation due to the amplitude imbalance \(\Delta A\) and phase imbalance \(\theta\) introduced by non-ideal factors. Taking into account the amplitude and phase imbalances, the IRR can be expressed as  
\[