## 1. Introduction

Biomedical electronics have acquired significant attention in healthcare, with a focus on the development of biosensors that enable online monitoring, detection, prevention, and personalized medicine for a variety of chronic and acute diseases. Especially in the last few years, there has been growing interest in the design of biomedical wireless sensors [

1,

2,

3]. Biomedical signals can be subdivided into two major classes: (1) endogenous signals that arise from natural physiological processes and are measured within or on living creatures (e.g., (EOG), electroencephalogram (EEG), electrocardiogram (ECG or EKG), electromyogram (EMG), temperature, blood glucose, etc.) and (2) exogenous signals applied from the outside (generally noninvasively) to measure internal structures and parameters. Endogenous bioelectric signals are invariably small, ranging from single microvolts to over 100 mV. Their bandwidths range from DC to perhaps 10 kHz at most [

4,

5,

6,

7]. The voltage and frequency ranges of some common biopotential signals are shown in

Figure 1.

A general biomedical system consists of an energy source, a differential amplifier, analog-to-digital conversion (ADC), digital signal preprocessing, and a communication subsystem. The ADC is one of the key building blocks, which enables converting analog signals from biomedical sensors to a digital format that can be easily processed and analyzed. For the design of the ADC, many authors choose the SARarchitecture due to its suitability for low-power and low-voltage requirements [

8,

9,

10,

11]. Recently, delta-sigma (

$\mathsf{\Delta}\mathsf{\Sigma}$) ADC has been gaining more and more popularity. Compared to other conversion techniques,

$\mathsf{\Delta}\mathsf{\Sigma}$ ADCs cover the widest conversion region of the resolution-versus-bandwidth plane, providing the most efficient solution to digitize diverse types of signals in many different applications such as biomedical ones [

12]. There exist two basic types of

$\mathsf{\Delta}\mathsf{\Sigma}$ modulators: discrete-time (DTDSM) and continuous-time (CTDSM). The DTDSM is more attractive for high-resolution applications due to its higher linearity and accuracy. On the other hand, less stringent amplifier speed specifications are required in CTDSM due to the absence of switches in the active-RC integrator, allowing achieving a higher speed of operation and lower power consumption. The asynchronous

$\mathsf{\Delta}\mathsf{\Sigma}$ modulator (ADSM) can be considered as a special type of CTDSM. ADSM is simple, does not require any clocking, matches well with mainstream CMOS technology, and can operate at low current and supply voltages [

13,

14,

15]. A comparison between DTDSM, CTDSM, and ADSM is shown in

Table 1.

In several publications, DTDSMs are used for biomedical signal processing [

1,

16,

17]. There also exists solutions utilizing ADSM [

18,

19,

20]. These ADSM are distinguished by very low power consumption in the order of tens of nanowatts. However, their bandwidth is very low in the order of tens of Hertz. The proposed ADSM covers the full bandwidth of endogenous bioelectric signals up to 10 kHz. The differential input range equals

${V}_{DDA}$ with a 0.5 V reference level (

${V}_{CM}$). The proposed ADSM with high resolution, wide bandwidth, and rail-to-rail input voltage range provides the universal solution for endogenous bioelectric signal processing. The circuit not only offers an alternative to the developed CTDSMs and DTDSMs, but it also fills the gap between published ADSMs, which do not allow processing the full spectrum of biomedical signals according to

Figure 1 except for those with a very high bandwidth in the order of MHz. An important parameter of ADSM is the center frequency, the calculation of which is part of this work. The following sections provide the details of our approach.

## 2. Asynchronous Delta-Sigma Modulator

There are two major types of architecture for $\mathsf{\Delta}\mathsf{\Sigma}$ modulators. The first one is the single-loop and the second the multi-loop architecture. Multi-loop architectures are commonly denoted as cascade or MASH (multi-stage noise shaping). A major drawback of MASH modulators is that precise matching of the analog and digital signal processing paths is required to avoid large errors (quantization noise leakage) caused by integrator gain coefficient variations. Because RC integrators are used in this design, where variations of about 20% in the $RC$ time constant can be expected, the single-loop architecture was chosen in this work. Its stronger ability to achieve high SNDR since it does not suffer from matching errors, which severely affect MASH modulators, is the major advantage in the design.

The block diagram of the second-order ADSM based on the cascade of integrators with distributed feedback (CIFB) topology is shown in

Figure 2.

The circuit consists of two integrators and a binary quantizer with hysteresis. The output

${V}_{OUTP}$ (or

${V}_{OUTN}$) is a pulse width modulated square wave of period

${T}_{PER}$ with a pulse width

${T}_{PW}$. The duty cycle

d is proportional to the amplitude of the input signal (Equation (

1)). Moreover, the period

${T}_{PER}$ of the asynchronous modulator output signal is modulated by the normalized input voltage

${V}_{IN}$ (Equation (

2)) [

21].

and:

where

${f}_{0}$ is the output carrier frequency,

${f}_{c}$ is the maximum value of

${f}_{0}$, namely the center frequency, and

$\left|{V}_{IN}\right|<1$ is the normalized input amplitude.

The center frequency of ADSMs determines the carrier-to-bandwidth ratio (

$CBR={f}_{c}/\left(2B\right)$, where

B is the input signal bandwidth), which is the ratio between the center frequency and the signal bandwidth. This ratio is equal to the oversampling ratio (

$OSR$) in synchronous delta-sigma modulators. It determines the minimal center frequency required for a certain conversion accuracy. The critical condition can occur when

${V}_{IN}$ is close to the full scale. The output frequency will decrease, and the high-frequency distortions around the center frequency shift to the low-frequency region. Consequently, distortions can leak into the baseband and adversely affect the modulator linearity for large input amplitudes. Therefore, the center frequency should be set far away from the baseband to avoid these components shifting into the signal baseband, and a high order filter is required to attenuate these out-band components. In order to achieve a high center frequency without degeneration of the linearity, the second-order topology was chosen. To calculate the center frequency of the proposed ADSM, the integrators’ output voltages are expressed as:

where

${I}_{1}\left(t\right)$,

${I}_{2}\left(t\right)$,

${I}_{3}\left(t\right)$, and

${I}_{4}\left(t\right)$ can be expressed as:

In order to find the center frequency, the timing diagram in

Figure 3 is considered, which corresponds to the schematic in

Figure 2.

The duty cycle of the ADSM is given by the ratio of rising (

${S}_{RE}$)-to-falling edge (

${S}_{FE}$) speed. To facilitate the equations, a symmetrical power supply is considered (

${V}_{DDAs}={V}_{DDA}-{V}_{CM}$,

${V}_{CM}$ = 0 V,

${V}_{SSAs}={V}_{SSA}-{V}_{CM},{V}_{REF}={V}_{DDAs}=\left|{V}_{SSAs}\right|;\left|{V}_{HL}-{V}_{CM}\right|={V}_{HH}-{V}_{CM}={V}_{H}$). During the

${T}_{1}$ period, the output voltage of the first integrator

${V}_{Y1}$ rises with speed, given by:

and the falling edge during

${T}_{2}$:

The first integrator output voltage swing is in the range of:

where

${V}_{H}$ is the comparator threshold voltage.

In order to find

${V}_{Y1\left(mean\right)}$, we calculate

${I}_{3\left(mean\right)}$. The duty cycle of the second integrator output

${V}_{Y2}$ is the same as the first one. From Equations (

9) and (

10), the value of

${I}_{3\left(mean\right)}$ is calculated to meet the duty cycle requirements.

For period

${T}_{1}$, we can write:

The entire period can be expressed as:

When a zero input is applied

${V}_{IN}=0$, the output of the ADSM is a square wave with a duty cycle of 50%. By defining

${T}_{C}$ as the period of the output signal, it can be calculated as:

Similar to the conventional synchronous CTDSMs, propagation delay is also an issue in ADSMs. The delay of the comparator increases the effective value of hysteresis and negligibly affects the center frequency of the ADSM. Therefore, the impact of the comparator delay,

$\tau $, on the center frequency of the proposed ADSM can be given by:

Equation (

17) shows that the center frequency

${f}_{C}$ of the modulator will decrease for a higher delay of the comparator, which degenerates the input bandwidth and linearity of the modulator [

20].

## 4. Simulation Results

The ADSM was designed utilizing the 0.18

$\mathsf{\mu}$m CMOS Logic or Mixed-Signal/RF, General Purpose process from the Taiwan Semiconductor Manufacturing Company. The circuit was designed for

${V}_{DD}$ = 1 V and

${I}_{BIAS}$ = 2.5

$\mathsf{\mu}$A. After completion of the layout design, its parasitic extraction was performed to find the parasitic resistances and capacitances corresponding to the designed devices and interconnects. After parasitic extraction, all simulations were performed using the Spectre simulator on the Cadence platform. The layout of the ADSM is shown in

Figure 9. The layout size is 350 × 155

$\mathsf{\mu}$m.

The ADSM output bitstream can be recovered by applying an ideal low pass filter with a cut-off frequency at the signal bandwidth. When ADSMs are used in A/D data conversion, a decoding circuit is required. The simplest one is the sample and hold circuit with a high sampling frequency. The time domain waveforms of the output signal

${V}_{OUTN}$ for

${V}_{IN}$ = 0 V and the corresponding frequency spectrum are shown in

Figure 10. The limit cycle frequency of the post-layout model of the ADSM was equal to 848 kHz and was very close to the calculated value in

Section 3.3 (

${f}_{C}=857$ kHz). The small difference was caused by the parasitic capacitances and resistances extracted from the layout.

Figure 11 shows the simulated spectrum of the ADSM for a sinusoidal input signal with an amplitude of (a) 100 mV (20% modulation depth) and (b) 500 mV (100% modulation depth). The corresponding spectra were obtained by applying a signal at

${f}_{IN}\le {f}_{Bandwidth}/3$ to include at least the second and third harmonic inside the band of interest. Due to this reason, the input frequency was set to 3.125 kHz, and then the third harmonic component was located in the 10 kHz bandwidth. The achieved

$SNDR$ was (a) 91.84 dB and (b) 78.13 dB. In the second case, the significant

$SNDR$ reduction was caused by higher harmonic tones.

Figure 12a shows the simulated dynamic range (

$DR$) with respect to the amplitude of the input signal with a frequency of both 3.125 kHz and 6.25 kHz. As the sine wave amplitude increased, the

$SNDR$ increased to reach the peak of 91.84 dB at −14 dBFS, and then dropped to 78.13 dB at 0 dBFS.

Figure 12b shows the

$SNDR$ vs. input signal frequency with an amplitude of both 100 mV and 500 mV. In the case of the input sine wave amplitude of 100 mV, the

$SNDR$ was above 88 dB for all frequencies up to the 10 kHz bandwidth. In the second case, the

$SNDR$ dropped from 110 dB to below 80 dB at frequencies smaller than 5 kHz (

${f}_{Bandwidth}/2$). This was because the second harmonic penetrated into the baseband and significantly reduced the

$SNDR$. The dynamic range was equal to 112 dB.

Table 4 presents a comparison of the proposed ADSM with other DSMs, which are capable of processing endogenous bioelectric signals in the full frequency range. Two figures of merit (

$FO{M}_{1}$,

$FO{M}_{2}$) are defined in Equations (

23) and (

24) for a better comparison. The first one emphasizes power consumption, whereas the second one emphasizes resolution. A better performance of DSMs is indicated by smaller

$FO{M}_{1}$ and larger

$FO{M}_{2}$ values.

As can be concluded from

Table 4, the proposed modulator offers a high

$SNDR$ and the best values of

$FO{M}_{2}$. Higher power consumption could be further improved utilizing a one-stage operational amplifier with much lower power consumption. The most attractive feature of ADSMs is the simple circuit architecture and clock-less operation. This feature can be very useful in applications in wireless sensors, where a decoding circuit (e.g., time-to-digital converter) is realized outside the integrated circuit.