## 4 Complex Continuous-Time Delta-Sigma ADCs: Principles and Challenges  
Complex bandpass continuous-time sigma-delta analog-to-digital converters (CT\(\Delta\)2 ADCs) offer sophisticated bandpass signal digitization, as shown in Figure 4[19]. These ADCs excel in software-defined radio and low-IF receiver applications by mitigating image problems without complex analog image-reject filters. The architecture uses a feedback-type quadrature bandpass continuous-time sigma-delta modulator (QB-CT\(\Delta\)2M), implementing complex integrators through cross-coupled I/Q integrator pairs [20]. The feedback topology delivers key advantages in the signal transfer function (STF): superior filtering characteristics, enhanced antialiasing without gain peaking, and elimination of the feedforward designs' summing stage before quantization [21]. This architecture effectively handles out-of-band blockers, which often exceed the desired in-band signal strength in receiver applications [22]. The sigma-delta loop's complex signal processing implements optimized transfer functions through careful noise transfer function design and strategic complex notch placement at the signal band center frequency [23]. This results in doubled noise-shaping capability over conventional real bandpass modulators. Figure 3: Blocker mask for the BLE Standard. Each QB-CT\(\Sigma\)AM stage comprises two cross-coupled low-pass CT\(\Sigma\)AM (LP-CT\(\Sigma\)AM) units. Figure 5 illustrates the block diagram of a single QB-CT\(\Sigma\)AM stage [24]. The single-loop LP-CT\(\Sigma\)AM filter transfer function H\({}_{\rm LP}\)(j\(\omega\)) is expressed in Equation (1) and depicted by the dotted line in Figure 5.  
\[\mathrm{H_{LP}(j\omega)}=\frac{1}{1+j\omega/\omega_{0}} \tag{1}\]  
The single-loop QB-CT\(\Sigma\)AM transfer function H\({}_{\rm BP}\) (jw) is derived by the frequency shifting from H\({}_{\rm LP}\)(j\(\omega\)) to center frequency \(\omega_{\rm C}\), substituting s-j\(\omega\)C for s. This shifts the LP-CT\(\Sigma\)AM cutoff frequency \(\omega_{0}\) to \(\omega_{\rm C}\pm\omega_{0}\). Equation (2) describes the resulting H\({}_{\rm BP}\)(jw), with Q representing the QB-CT\(\Sigma\)AM quality factor. \[\mathrm{H_{BP}(j\omega)}=\mathrm{H_{LP}(j\omega-j\omega_{C})}=\frac{1}{1+j \omega/\omega_{0}-j\omega_{C}/\omega_{0}} \tag{2}\]  
The implementation of an ADC with QB-CT\(\Sigma\)AMs presents several critical challenges that must be carefully addressed. Path mismatches between I/Q channels can significantly degrade image rejection performance, necessitating careful layout techniques and, potentially, calibration mechanisms [25]. Clock jitter sensitivity is another crucial concern, as timing errors in the feedback DAC directly impact conversion accuracy, requiring robust clock distribution networks and carefully designed DAC switching schemes [26]. The feedback topology, which must process both the input signal and noise, demands appropriate filter coefficient selection and circuit topology optimization to accommodate larger  
Figure 4: Block diagram of complex bandpass continuous-time delta–sigma modulator. Figure 5: Block diagram and transfer functions H (j\(\omega\)) of QBP-CT\(\Sigma\)AM. signal swings effectively. Stability considerations become particularly complex due to interactions between I/Q paths, requiring comprehensive stability analysis that accounts for variations in operating conditions [27]. Additionally, various non-idealities--such as I/Q mismatches, DAC element mismatching, and excess loop delay--often necessitate sophisticated calibration schemes [28]. Despite these challenges, the continuous-time implementation offers several advantages over discrete-time alternatives, including inherent antialiasing filtering, lower power consumption, higher potential sampling rates, and freedom from switched-capacitor settling constraints [29].