A 22.3-Bit Third-Order Delta-Sigma Modulator for EEG Signal Acquisition Systems

Abstract

This paper presents a high resolution delta-sigma modulator for continuous acquisition of electroencephalography (EEG) signals. The third-order single-loop architecture with a 1-bit quantizer is adopted to achieve 22.3-bit resolution. The effects of thermal noise on the performance of the delta-sigma modulator are analyzed to reasonably allocate the switched-capacitor sizes for optimal signal to noise ratio (SNR) and minimum chip area. The coefficients in feedback path and input path are optimized to avoid the signal distortion under the full-scale input voltage range with almost no increase in total capacitance sizes. Fabricated in 0.5 µm CMOS technology and powered by a 5 V voltage supply, the proposed delta-sigma modulator can achieve 136 dB peak SNR with 16 Hz input and 137 dB dynamic range in 100 Hz signal bandwidth with an oversampling ratio of 512. The modulator dissipates 700 µA. The core chip area is 1.96 mm${}^2$. The modulator occupies 1.41 mm${}^2$ and the decimator occupies 0.55 mm${}^2$.

1. Introduction

The monitoring of biopotential signals is a key and important component of the medical diagnostic system. Electroencephalography (EEG) waves are currently one of the most commonly used biopotential signals for monitoring brain activity [1]. In the analog-front-end (AFE) circuits of the whole system, ADCs are used to collect and digitize EEG signals. However, there are some challenges in ADC design. First, the EEG signals have extremely low frequency behavior and low signal amplitude (i.e., 0.5–70 Hz and 1–100 µV for surface EEG [2]). They are very susceptible to various aggressors, such as 1/f noise from CMOS transistors, power line interference, and thermal noise [3]. Second, dry electrodes with high impedance are preferred for continuous monitoring of brain signals [4], which will result in weaker EEG signal and lower common-mode-rejection-ratio (CMRR) due to mismatches. Additionally, charge accumulation caused by the natural humidity of dry electrodes can lead to the electrode-offset-voltage (EOV) of ±200–300 mV [5]. The large EOV will result in saturation of the integrator so as to affect the resolution of the modulator. Therefore, to achieve stable and accurate measurement of EEG signals, high resolution and large input voltage range ADCs are required.

Compared with other ADC structures, the delta-sigma ADC is promising for the application to convert EEG signals with higher resolution performance in the low frequency range [6]. Oversampling technology is employed to reduce quantization noise and noise shaping technology is used to push the noise to higher frequency bands. Then, a digital decimation filter is used to filter out the shaped noise in the high frequency domain, so as to achieve high conversion resolution. Due to the low power consumption and high integration, the delta-sigma ADCs have also proven their practicality in biomedical applications [7]. For example, a continuous-time delta-sigma modulator using a modified instrumentation amplifier and current reuse DAC for neural recording is proposed in [8], with the peak signal to noise and distortion ratio (SNDR) of 78 dB and the dynamic range (DR) of 90 dB are achieved for a bandwidth of 250 Hz. In [9], a low-power delta-sigma modulator with a low-noise amplifier in the first integrator is proposed, which can achieve a measured 96 dB DR, over a 250 Hz signal bandwidth, with an oversampling ratio (OSR) of 500.

There are three common ways to boost the SNR and SNDR of a delta-sigma modulator. First, increase the OSR with higher sampling frequency $f_s$. But this will result in more power consumption. Second, increase the order of the loop filter in the single-loop structure to achieve more effective noise shaping (NS). But this will increase the circuit complexity and degrade the loop stability. Compared with the single-loop structure, the multistage noise-shaping (MASH) can ensure loop stability with higher NS order [10,11]. However, due to PVT variations, the mismatch of the transfer function between analog and digital domains can lead to quantization-error leakage and reduce conversion resolution. The third method is to increase the resolution of the quantizer, but multi-bit architecture requires the dynamic element matching (DEM) technique, which also requires extra current consumption for the extra blocks and increases the circuit complexity.

Taking the above conditions into consideration, this paper presents a single-loop third-order structure with a single-bit quantizer to achieve a delta-sigma modulator with 136 dB SNR for EEG monitoring. In addition, the thermal noise effects are analyzed for minimum total capacitance and the modulator coefficients are optimized to avoid distortion of the integrator in the full-scale input range. As a result, the proposed modulator has a higher overload level, which is good for expanding the dynamic range in the EEG acquisition system.

The detail of the proposed modulator topology design is presented in Section 2. Section 3 describes the circuit implementation details. Section 4 presents the simulation and measurement results and is followed by the conclusion.

2. Thermal Noise Analysis and Proposed Topology Design

In this section, topology architectural design of the proposed high-resolution delta-sigma modulator is discussed. It is essential to analyze the thermal noise effects in the delta-sigma loop before the topology design, so as to properly allocate the switched-capacitor sizes for optimal SNR and minimum chip area. Considering the second-order modulator loop using the feedforward topology shown in Figure 1 and Figure 2, we shall find the minimum acceptable capacitor sizes that satisfy a specified SNR by the following steps.

Figure 1. Noise sources in the feedforward topology.

Figure 2. Schematic of the feedforward topology.

$\left(1\right)$ Identify the main thermal noise sources of the modulator circuit, including the first integrator, the second integrator, and the adder circuit in front of the quantizer.

$\left(2\right)$ Find the PSD of the noise generated by each thermal noise source. The mean-square $\left(MS\right)$ value of the input-referred noise voltage of the first integrator can be expressed as

$$\overline{v_{ni1}^2}=\frac{KT}{C_{s1}}·\frac{7/3+2x}{1+x}$$

(1)

Here, the parameter x = $2R_{on}{g}_{m1}$. The noise power at the input of the second integrator can be expressed as

$$\overline{v_{ni2}^2}=\frac{KT}{C_{s2}}·\frac{7/3+2x}{1+x}$$

(2)

In the adder circuit, the noise power of the three switched-capacitor branches can be equivalently represented as

$$\begin{array}{c}\hfill \overline{v_{no2}^2}=\frac{2KT}{C_{f1}}·(1+\frac{C_{f2}}{C_{f1}}+\frac{C_{f3}}{C_{f1}})\\ \hfill =\frac{2KT}{C_{f1}}(1+2+1)\end{array}$$

(3)

Here, $C_{f1}$, $C_{f2}$ and $C_{f3}$ are ratioed. Since all these sources produce sampled white noise, the PSD of the jth source can be given by

$$S_{vj}=\frac{\overline{v_j^2}}{f_s/2}$$

(4)

$\left(3\right)$ Find the noise transfer function (NTF) from each noise source to the output. The integrator block can be denoted by

$$H\left(z\right)=\frac{z^{-1}}{1-z^{-1}}$$

(5)

The NTF from the input of the first integrator to the output of the modulator is given by

$$NT{F}_{i1}\left(z\right)=\frac{H^2+2H}{1+2H+H^2}=2z^{-1}-z^{-2}$$

(6)

The NTF from the input of the second integrator to the output is given by

$$NT{F}_{i2}\left(z\right)=\frac{H}{1+2H+H^2}=z^{-1}(1-z^{-1})$$

(7)

The NTF from the output of the second integrator to the output of the modulator is given by

$$NT{F}_{o2}\left(z\right)=\frac{1}{1+2H+H^2}={(1-z^{-1})}^2$$

(8)

$\left(4\right)$ Integrate each noise PSD in the frequency band of interest. If OSR≫ 1, the resulting output noise powers are

$$\begin{array}{c}\hfill \overline{N_{i1}^2}=\frac{\overline{2V_{ni1}^2}}{f_s}{\int }_0^{f_s/\left(2OSR\right)}\mid {NTF}_{i1}\mid df\\ \hfill =\overline{V_{ni1}^2}[\frac{5}{OSR}-\frac{4}{\pi }sin\left(\frac{\pi }{OSR}\right)]\approx \frac{\overline{V_{ni1}^2}}{OSR}\end{array}$$

(9)

$$\overline{N_{i2}^2}=\overline{V_{ni2}^2}[\frac{2}{OSR}-\frac{2}{\pi }sin\left(\frac{\pi }{OSR}\right)]\approx \overline{V_{ni2}^2}\frac{{\pi }^2}{3OS{R}^3}$$

(10)

$$\overline{N_{o2}^2}=\overline{V_{no2}^2}[\frac{6}{OSR}-\frac{2}{\pi }sin\left(\frac{\pi }{OSR}\right)cos\left(\frac{\pi }{OSR}\right)-\frac{8}{\pi }sin\left(\frac{\pi }{OSR}\right)]\approx \overline{V_{no2}^2}\frac{{\pi }^4}{5OS{R}^5}$$

(11)

$\left(5\right)$ Calculate the sum of all these contributions. With x = 0.1 in the expressions for $\overline{V_{ni1}^2}$ and $\overline{V_{ni2}^2}$, the total output thermal noise power is

$$\begin{array}{c}\overline{N_{total}^2}=\overline{N_{i1}^2}+\overline{N_{i2}^2}+\overline{N_{o2}^2}\\ =\frac{2.3}{OSR}·\frac{KT}{C_{s1}}+\frac{7.56}{OS{R}^3}·\frac{KT}{C_{s2}}+\frac{155}{OS{R}^5}·\frac{KT}{C_{f1}}\end{array}$$

(12)

With OSR = 512 in this design, $\overline{N_{total}^2}$ the can be simplified as

$$\overline{N_{total}^2}=4.5\times {10}^{-3}\frac{KT}{C_{s1}}+5.6\times {10}^{-8}\frac{KT}{C_{s2}}+4.4\times {10}^{-12}\frac{KT}{C_{f1}}$$

(13)

This expression shows that $C_{s1}$ in the first integrator dominates the total output thermal noise power; $C_{s2}$ and $C_{f1}$ can be ignored because the noise of these capacitors is shaped by first-order and second-order high-pass transfer functions.

After the above analysis, an important optimization can be performed to minimize the sampling capacitance for a desirable thermal noise level according to

$$SNR=\frac{1}{OSR}·\left(\frac{V_{in}^2/2}{8KT/C_{s1}}\right)$$

(14)

where $V_{in}$ is the amplitude of a half-scale signal. The capacitors in the other switched-capacitor structures can be chosen to be much smaller than $C_{s1}$.

Figure 3 shows the optimized single-loop third-order delta-sigma modulator topology. The integrator coefficients $c_i$ and feedforward coefficients $a_i$ are optimized with significant benefits in terms of silicon area and load capacitance. The feedforward paths are all summed at the input node of a comparator. Values $k_1$ and $k_2$ are used to amplify the feedback and input signals, so that the detected input signal range of the modulator can reach full scale. In order to avoid the signal distortion at the output of the first integrator, the amplified difference between input and feedback signals is scaled down by $k_3$. Therefore, with $k_1$, $k_2$, and $k_3$, the proposed delta-sigma modulator can achieve high SNR over the full-scale input range of 0 V to 5 V with almost no increase in total capacitance sizes. The NTF of the designed modulator can be calculated as

$$NTF\left(z\right)=\frac{{(z-1)}^3}{{(z-1)}^3+k_1{k}_3{a}_1{c}_1{(z-1)}^2+k_1{k}_3{a}_2{c}_1{c}_2(z-1)+k_1{k}_3{a}_3{c}_1{c}_2{c}_3}$$

(15)

The coefficients of the designed modulator are summarized in Table 1. The ideal output spectrum, obtained with a behavioral simulation, is shown in Figure 4. With OSR of 512, the simulated SNDR at 16 Hz is 145.8 dB.

Figure 3. Proposed single-loop third-order modulator topology.

Table 1. Coefficients of the designed modulator.

Feed-Forward Coefficients	Integrator Coefficients	Other Coefficients
a${}_1$ = 2	b${}_1$ = 1/3	k${}_1$ = 2
a${}_2$ = 3	b${}_2$ = 1/4	k${}_2$ = 2
a${}_3$ = 4	b${}_3$ = 1/16	k${}_3$ = 1/2

Figure 4. Ideal output spectrum with a 16 Hz input signal.

3. Circuit Implementation

This section describes some of the issues that need to be considered before circuit design, such as the integrator output swings, the OTA requirements, and the single-bit quantizer design. Then, the complete circuit-level implementation of the proposed modulator is explained.

3.1. Integrator Output Swings

As the proposed modulator is optimized to avoid signal distortion under full-scale input signal range, it is necessary to simulate the output voltage swing of each integrator. The behavioral simulation is performed with a full-scale input, and the output swing of each integrator after the input normalized to the reference voltage is shown in Figure 5. As we can see in this figure, the output swing of the first integrator is the largest, but it still does not exceed 60% of the reference voltage, which means that there will be no signal distortion with full-scale input signal.

Figure 5. Integrator output swings with a 16 Hz full-scale input signal. (a) Output of the first integrator. (b) Output of the second integrator. (c) Output of the third integrator.

3.2. OTA Requirements

The OTA is a key component of the delta-sigma modulator and determines its main performance. The requirements for OTA mainly include DC gain, output swing, and gain bandwidth (GBW). The output swing determines the maximum input amplitude $V_{in,max}$, which is very important for the dynamic range (DR) of a KT/C noise dominated modulator.

$$DR=\frac{P_{in,max}}{P_{KT/C}}=\frac{V_{in,max}^2·OSR·C_S}{2KT}$$

(16)

Therefore, a larger output swing allows for a reduction in sampling capacitance, thus reducing power consumption and chip area. In addition, the OTA DC gain must have sufficient margin for the process variations to avoid circuit performance fluctuations of the modulator. Figure 6 shows the relationship between SNR and OTA gains in the proposed topology. We can see that if the gain is above 50 dB, the wide variation does not affect the SNR significantly. The SNR starts to fall bellow 130 dB when the OTA gain is lower than 30 dB. But this gain requirement only takes into account noise shaping; higher OTA gain is required for better distortion performance in practical applications.

Figure 6. The SNR versus the OTA gains.

A typical full-differential two-stage amplifier with a common-mode feedback (CMFB) circuit is employed in the modulator to achieve guaranteed circuit performance. The schematic is shown in Figure 7. Transistors PM0, NM0, and PM5 form the amplifier’s first stage to provide high DC gain. PM3 and NM3 form the second stage to provide large output voltage swing. PM4, NM4, and PM6 form the CMFB circuit to sense the output common-mode voltage and provide the control voltage to balance the OTA’s positive and negative outputs. R1A and R1B are the feedback resistors. R0 and C0 form the Miller compensation for stabilizing the OTA.

Figure 7. Schematic of the OTA in the modulator.

It should be noted that the thermal noise and flicker noise of the first OTA is not shaped by the loop filter, and it directly influences the modulator performance. Particularly, the 1/f noise is dominant in very low frequency. Therefore, the input devices PM0A and PM0B, the main noise contributors, are designed with large dimension and large $g_m$ in order to suppress the OTA noise below the modulator overall noise floor. The other OTAs are relatively less critical than the first OTA; therefore, the current and performance are scaled down to reduce power consumption. From the post-layout typical simulation, the first OTA achieves 108 dB DC gain, 55° phase margin, and 13 MHz GBW with a 100 pF load. The other two OTAs achieve 98 dB DC gain, 12.5 MHz GBW, and 65° phase margin with a 100 pF load. The first and other OTAs draw 340$\mathsf{\mu }$A and 100$\mathsf{\mu }$A, respectively.

3.3. Single-Bit Quantizer Circuit

Figure 8 shows the single-bit quantizer circuit, including a dynamic comparator and an SR latch. At phase ${\varphi }_1$, the input of the comparator is reset to $V_{CM}$. Then, the differential signals at the summation node $N_{sum}$ in Figure 3 are compared at phase ${\varphi }_2$. Two equal capacitors are connected in parallel, and the plates of $C_1$ and $C_2$ are opposite in order to achieve matching of the mim capacitors.

Figure 8. Schematic of the single-bit quantizer.

3.4. Complete Modulator Circuit Implementation

Figure 9 shows the complete circuit implementation of the proposed modulator. In order to improve the common mode noise rejection and avoid performance degradation, a fully differential circuit topology is adopted throughout the design [12]. The input signal is sampled during ${\varphi }_1$; the voltage difference between the input and $V_{cm}$ is stored in the sampling capacitors. Depending on the digital output of the quantizer, Lp or Ln will be opened and feedback reference levels will be applied to the sampling capacitors, which will perform the subtraction operation and then transfer charge from the sampling capacitors to the integration capacitors to complete the integration function during ${\varphi }_2$. The gain coefficients of the proposed modulator in Figure 3 are easily realized by appropriate capacitance sizes.

Figure 9. Schematic of the presented third-order modulator.

4. Measurement Results

The designed modulator is fabricated in a 0.5 $\mathsf{\mu }$m standard CMOS process. Only regular threshold MOS transistors are adopted in this design. The chip microphotograph is shown in Figure 10. The core chip area is 1.96 mm${}^2$. The modulator circuit occupies 1.41 mm${}^2$ and the decimator occupies 0.55 mm${}^2$.

Figure 10. Chip photograph.

Figure 11 shows the measured output spectrum with 16 Hz input after decimation; the number of samples used for the PSD plot is 16,384. Figure 12 shows the noise floor of the proposed modulator. With the full-scale input of 5 Vpp, the measured ENOB is 22.3 bit. Figure 13 shows the measured SNR versus the input signal amplitudes normalized by reference voltage. The peak SNR reaches 136 dB, and the dynamic range is 137 dB in a 100 Hz signal bandwidth. The modulator consumes 700 $\mathsf{\mu }$A and the decimator consumes 120 $\mathsf{\mu }$A. Table 2 gives the summary of the measurement results.

Figure 11. Measured output spectrum with 16 Hz input.

Figure 12. Noise floor of the proposed modulator.

Figure 13. Measured SNR versus the input signal amplitudes normalized by reference voltage.

Table 2. Measurement results.

Supply voltage	$3.3$$\sim$5 V
Analog power consumption	$700\mathsf{\mu }$A
Peak SNR	136 dB
Dynamic range	137 dB
Sampling frequency	$102.4$ kHz
Signal bandwidth	100 Hz
Oversampling ratio	512
Modulator core size	1.41 mm${}^2$
Figure of merit	188

The performance of our modulator is compared with other published high resolution modulators in Table 3. The common FOM equation [13] is shown below:

$$FOM=D{R}_{db}+10log\left(\frac{BW}{P}\right)$$

(17)

where $D{R}_{db}$ is the dynamic range of the modulator, BW is the signal bandwidth, and P is the power consumption. As we can see in Table 3, our modulator shows the highest FOM and is suitable for application of an EEG acquisition system.

Table 3. Performance comparation.

References	2003 [14]	2012 [15]	2015 [16]	2019 [17]	2019 [8]	2022 [9]	This Work
Techn. [nm]	350	700	180	65	180	90	500
Supply [V]	2.6	5	5	1	1.8	1.2	3.3∼5
Arch.	DT-MB	CT-MB	DT-SB	CT-SB	CT-SB	DT-SB	DT-SB
BW [Hz]	45	10	100	150	250	250	100
ENOB [b]	14.7	16.7	16.4	12.2	12.7	14.8	22.3
DR [dB]	98	121	110.1	99.3	90	95.6	137
Cons. [$\mathsf{\mu }$A]	23	240	101	20.8	12.8	25	700
Area. [mm${}^2$]	0.7	N.D	0.8	0.225	0.088	0.39	1.41
S.FOM [dB]	156.8	160.2	163.1	167.8	153.1	164.8	188

5. Conclusions

In this paper, a high resolution delta-sigma modulator for acquiring and digitization of EEG signals is presented. A third-order single-loop modulator architecture with a 1-bit quantizer is adopted. By analysing the thermal noise effects and optimizing the feedback and input coefficients, the proposed modulator can achieve best SNR in a full-scale input range of 0 V to 5 V with almost no increase in total capacitance sizes. Fabricated in 0.5 $\mathsf{\mu }$m CMOS technology and operated at 5 V voltage supply, the proposed modulator can achieve 136 dB peak SNR and 137 dB DR with 100 Hz bandwidth. Combined with the characteristics of EEG signals, the measurement results show that this modulator can be used for the application of the EEG acquisition system.