## 2 Materials and Methods
### Synchronous Filtering Concept  
Synchronous filtering is a widely used technique for carrier recovery and image rejection in radio-frequency signal processing [46]. A simplified diagram of a synchronous filter is shown in Figure 2a. It consists of a core filter enclosed by mixers [47]. Each mixer  
Figure 1: Biopotential readout circuits for synchronous filtering with (**a**) and without (**b**) a driven reference electrode (DRE) module. is a precision multiplier. The first one (M1) is a demodulator, the second one (M2) is a modulator, and the net result of the SF system behaves as an equivalent filter without mixers. Synchronous filtering has the functionality to accomplish some typically difficult tasks for design filters (i.e., very high-frequency, very high-Q band-pass filters) with a relatively simple core filter design [47]. The SF operation can be simply explained with the following example. Let us assume that the reference signal (\(V_{REF}\)) is in phase with the input signal (\(V_{SIG}\)) and both are sinusoidal waveforms, as written in Equations (1) and (2):  
\[V_{SIG}=V_{s}\sin(\omega\mathrm{t}) \tag{1}\]  
\[V_{REF}=V_{r}\sin(\omega\mathrm{t}) \tag{2}\]  
The output of the first mixer \(M1\) is the product of (1) and (2), expressed by (3):  
\[V_{M1}=0.5V_{s}V_{r}-0.5V_{s}V_{r}\cos(2\omega\mathrm{t}) \tag{3}\]  
The core filter is usually a low-pass filter (LPF) with a sufficiently low cutoff frequency (\(f_{3 dB}\)); therefore, it passes the DC component and filters out the AC component with doubled frequency in (3). Thus, the SF output signal becomes:  
\[V_{SF}=0.5V_{s}V_{r}^{2}\sin(\omega\mathrm{t})=GV_{s}\sin(\omega\mathrm{t}), \tag{4}\]  
where the gain \(G\) of the SF algorithm is expressed by (5):  
\[G=0.5V_{r}^{2} \tag{5}\]  
When \(V_{r}=\) /2 = 1.414 V, the SF algorithm recovers the input signal \(V_{SIG}\) with unity gain. The second mixer M2 is an up-converter and translates the cutoff frequency \(f_{3 dB}\) of the core filter in two sidebands around the \(V_{REF}\) frequency (\(f_{nr}\)). Thus, the SF filter acts as a band-pass filter with a quality factor Q, given by (6):  
\[\mathrm{Q}=\frac{f_{ref}}{2f_{3 dB}} \tag{6}\]  
It can be deduced from (6) that a very high Q factor can be achieved by configuring the cutoff frequency \(f_{3 dB}\) and its ratio to \(f_{nr}\). SF algorithms have great advantages in carrier extraction and recovery. For example, let us assume that \(V_{SIG}\) is disturbed by a spike noise and occasionally a few periods are missing. Due to the large LPF time constant, the filtered carrier (\(V_{SF}\)) can be recovered and free of such spike noise disturbance. In order to operate with arbitrary phase differences between \(V_{SIG}\) and \(V_{REF}\), an orthogonal SF architecture is  
Figure 2: Diagram of the synchronous filtering concept: (**a**) in-phase SF, (**b**) out-of-phase SF design with a quadrature demodulator-remodulator. conventionally used [48]. It includes two parallel SFs shown in Figure 2a, operating with quadrature reference signals. Their outputs are summed, as shown in Figure 2b. The left mixers M1 and M3 are quadrature demodulators, while the right mixers M2 and M4 are quadrature remodulators, and together all of them perform two frequency conversions. First, the spectrum of the input signal is shifted from the two sidebands around \(f_{\textit{rpf}}\) to the baseband, i.e., close to DC. After low-pass filtering of the demodulated spectral components, they are shifted again from the baseband to the two sidebands around \(f_{\textit{rpf}}\), i.e., the spectrum is restored into the two sidebands, but the bandwidths are limited to \(f_{\textit{rpf}}\pm f_{\textit{3 dB}}\). Thus, the quadrature SF architecture, employing quadrature demodulator-remodulator (QDR), exhibits a band-pass frequency response. The QDR architecture was introduced about 70 years ago by [PERSON] for the generation and detection of single-sideband radio signals [48]. We should point out some important properties of the QDR architecture.