## 5 Proposed Quadrature Bandpass Continuous-Time Delta-Sigma ADC
### Quadrature Bandpass Continuous-Time Delta-Sigma ADC Circuit Design  
#### 5.2.1 Four-Bit Feedback Current Digital-to-Analog Converter Circuit  
Four-bit feedback digital-to-analog converters (FBDACs) are circuits that implement the negative feedback factors a\({}_{1}\), a\({}_{2}\), a\({}_{3}\), and g\({}_{1}\) of the QBP-CT\(\Delta\)M, as shown in Figure 6. They employ the bipolar topology depicted in Figure 7b rather than the unipolar approach shown in Figure 7a, which allows the DAC current to be reduced by half compared to the unipolar version. To improve the output impedance of the DAC unit cell, cascode transistors are used in the current source, utilizing both NMOS and PMOS configurations. In the current design, the unit current of the bipolar topology is specifically designed to be half that of the unipolar topology. This design choice serves two purposes: reducing noise contribution and lowering current consumption in the bipolar topology. Additionally, it enables smaller capacitor loads, resulting in a more compact overall design. The linearity of the DAC is critical for the QBP-CT\(\Delta\)M to achieve the required SNR. To address DAC linearity issues, a data-weighted averaging (DWA) technique is implemented, effectively mitigating these concerns without requiring large device areas for intrinsic device size matching. The DWA system operates by updating the pointer location in each cycle based on the current position and incoming data, as shown in Figure 7c. The pointer rotation process begins with the initialization of a pointer, typically set to zero, which serves as the starting point for element selection. When a digital input sample enters the DAC, it undergoes conversion into a thermometer code, representing the number of elements that need to be activated for that particular input. The DWA algorithm then proceeds to select DAC elements sequentially, starting from the current pointer position. It activates as many elements as required by the input code, ensuring that each element contributes equally over time. After processing the input sample, the pointer is updated  
Figure 7: (**a**) Unipolar and (**b**) bipolar current DAC cells and (**c**) 4-bit DWA pointer rotation. to indicate the next unused element, preparing the system for the subsequent input. This continuous update mechanism is crucial for maintaining the balanced usage of all DAC elements. In cases where the pointer reaches the end of the available DAC elements, it employs a wraparound technique, returning to the beginning of the element array. This wraparound ensures that the rotation process continues seamlessly, maintaining the equal utilization of all elements over extended periods of operation. By consistently rotating through all available elements, the DWA pointer rotation process effectively distributes any mismatch errors across the entire DAC, resulting in improved overall linearity and performance. Notably, the pointer-shifting logic for the DWA is kept separate from the main feedback path, ensuring that the pointer for the current data is immediately available without introducing additional logic delays. As a result, the only delay in the system comes from data propagation through the matrix. Figure 8 shows Monte Carlo simulation results comparing the performance of a complex bandpass sigma-delta ADC with and without DWA. The results are presented in two histograms: one for spurious-free dynamic range (SFDR) and one for IRR. In the SFDR analysis, the two configurations show distinct characteristics. Without DWA, the system achieves a mean of \(-\)75.1 dB with a standard deviation of 1.1 dB. When DWA is implemented, the performance improves to a mean of 83.2 dB with a standard deviation of 1.2 dB, demonstrating significant enhancement in spurious performance. The distribution of SFDR values spans from approximately 72 dB to 85 dB, showing consistent performance across multiple simulation runs. For the IRR measurements, there is a notable difference between the two configurations.