# A Fully Integrated Low-Power Multi-Mode RF Receiver for BDS-3/GPS
## 4 Receiver Design
### CBPF  
The signal after GMT is centered at 4.092 MHz, and BDS-3/GPS signals are buried below the thermal noise. For BDS-3/GPS, the required bandwidth is 4.092/2.046 MHz, respectively. In order to realize a complex filter, a third-order real low-pass filter needs to be realized first, and then the IF shift is realized by quadrature coupling [24]. In our design, taking the third-order RLC low-pass filter as the prototype, a third-order Butterworth-type in-phase and quadrature (IQ) CBPF is realized by leapfrog. There are DC-blocking capacitors at the output of the Gm, so the RF common-mode (CM) voltage is not transmitted to the mixer. In addition, there is a common-mode feedback (CMFB) circuit in the Tia. Therefore, the CM voltage of the Tia is the DC operating point of the mixer. Moreover, the CMFB circuit of the later-stage CBPF is the same as that of Tia. So the CM voltages of these two sub-circuits are the same. There is no need for blocking capacitors between GMT and CBPF. Moreover, compared with the traditional method of adding DC-blocking capacitors to the CBPF input, this method uses fewer capacitors and a smaller capacitance value, thus conserving space. RC calibration is employed to adjust the bandwidth and IF digitally [25] to avoid suffering from process changes. Under different simulation corners, the resistance and capacitance values vary by \(\pm\)15% and \(\pm\)10%, respectively. This will cause a deviation of \(-\)23.5%\(\sim\)+26.5% in the RC time constant. Thus a deviation of \(-\)20.95%\(\sim\)+30.72% in the center frequency will be observed without the RC calibration. Such a large deviation will cause useful signals to be greatly suppressed. To ensure that the bandwidth of CBPF can  
Figure 3: RF front-end circuit where \(L_{1}\) and \(L_{2}\) represented by dashed line are realized by bonding wires. cover the useful signals without loss under the process, voltage, and temperature (PVT) variations, the center frequency deviation range should be controlled within \(\pm 2\%\), so the RC calibration accuracy should be within the range of \(-1.96\%\)\(\sim\)\(+2.04\%\). To reserve a certain margin, the RC calibration accuracy is set within \(\pm 1.8\%\) in this design. The RC calibration diagram is shown in Figure 4. Firstly, the RST signal is enabled to discharge the calibration capacitor \(C_{cal}\) until its voltage \(V_{c}\) equals zero. Then RST is disabled, and the signal EN is enabled and lasts \(T_{C}\). The reference current \(I_{ref}\) is converted from the reference voltage \(V_{ref}\) and is equal to \(V_{ref}/R_{ref}\). After a current mirror, this current is used to charge \(C_{cal}\) when EN is enabled. \(C_{cal}\) is a replica of the capacitor in CBPF. The charged voltage \(V_{c}\) is compared with \(V_{ref}\) to generate the signal C[5:0] controlling \(C_{cal}\). At the calibration beginning, \(C_{cal}\) is the smallest and then increases gradually. A process--such as the discharge, charge, and comparison--is required for each increase. In the beginning, \(V_{c}\) exceeds \(V_{ref}\) within \(T_{C}\), making the signal 'ampout' low. As \(C_{cal}\) increases, the \(V_{c}\) obtained within TC becomes smaller until it is equal to \(V_{ref}\). At this point, the 'ampout' reverses to high and the calibration concludes, holding (3). \[R_{ref}\,C_{cal}=T_{C}, \tag{3}\]  
With C[5:0] obtained at this point, the desired RC is achieved. There is a mismatch in the current mirror, which is represented by \(\alpha\) in Figure 4. There is a DC offset voltage, \(V_{off}\), of the comparator. Both factors will introduce errors to the RC time constant, and the error can be expressed as  
\[RC_{\rm error}=\frac{\alpha V_{ref}-V_{\rm off}}{V_{ref}+V_{off}}\,T_{c} \approx\,\left(\alpha-\frac{V_{\rm off}}{V_{ref}}\right)\,T_{C}, \tag{4}\]  
As mentioned above, the simulated deviation in the RC time constant is \(-23.5\%\)\(\sim\)\(+26.5\%\). In the design, the change of \(C_{cal}\) is considered as \(\pm 30\%\).