A 0.5 V, 32 nW Compact Inverter-Based All-Filtering Response Modes Gm-C Filter for Bio-Signal Processing

Abstract

A low-power, low-voltage universal multi-mode Gm-C filter using a 180 nm TSMC technology node is presented in this paper. The proposed filter employs only three transconductance operational amplifiers (OTAs) operating in the sub-threshold region with a supply voltage of 0.5 V, resulting in a power consumption of 32 nW. Moreover, without additional active elements, the proposed circuit can operate various functional modes, such as voltage, current, transconductance, and trans-resistance. The filter’s frequency, centered at 462 Hz, and a compact and low-power solution showing only 93.5 µVrms input-referred noise make the proposed filter highly suitable for bio-signal processing.

1. Introduction

Nowadays, low-power techniques in integrated circuits (ICs) design have gained a critical role in low-power application systems. Several advanced approaches have been employed to reduce power consumption, including lowering the supply voltage, bulk-driven techniques, floating gates, and biasing transistors in the subthreshold region [1,2,3]. Among these, reducing the supply voltage directly reduces energy consumption, extending operation within a given power budget [3,4,5,6,7,8]. In any case, the most extreme care ought to be given to not jeopardize the proper operation of the circuit, characterized at the application level. This is particularly important for low-power bio-signal sensing devices in biomedical applications that are frequently combined with devices that gather and store energy [9,10,11]. However, if a high supply voltage is required, the system should be equipped with DC/DC or AC/DC converters, depending on the energy available source type [12,13]. When the converters are used in energy harvester systems, the conversion efficiency can be estimated in the range between 40% and 80%, leading to power waste [14]. As a consequence, to improve the performance and efficiency of an energy-harvested system, the use of low-power techniques will be fundamental [15,16,17].

From the perspective of making sensory devices, integrated filter blocks exhibit advantageous features, such as signal conditioning capabilities, and the elimination of interference and noise. One of the common filter topologies is Active-RC, which is usually used in communication systems [18]. Despite its good accuracy and low distortion, it cannot be used in applications that require low power. In analog integrated circuits, Gm-C filters are among the main building blocks, and their use has led to excellent performance, both in terms of chip size and power consumption [19,20,21,22]. An additional way to further reduce the power consumption of Gm-C filters is to implement their operational transconductance amplifiers (OTAs) by using inverter-based topologies. In fact, inverter-based OTAs offer supply voltage scalability, and thus are very effective at reducing power consumption. A range of ultra-low power analog filters with inverter-based topologies has been described in [23,24,25], which feature a high-frequency response and low power consumption. For different applications, analog filters with different frequency responses are required, including low-pass (LP), high-pass (HP), band-reject (BR), all-pass (AP), and band-pass (BP). Therefore, the design of a universal filter capable of generating all possible filtering responses is often required [26,27]. There are several modes of operation for multi-mode analog filters, such as voltage mode (VM), current mode (CM), trans-resistance mode (TRM), and transconductance mode (TCM). This paper describes a low-power integrated Gm-C filter capable of generating all filtering responses under the respective operation modes.

This paper is structured as follows: Section 2 outlines the proposed filter design, while Section 3 presents the simulation results. Section 4 details the noise analysis, and Section 5 covers the sensitivity analysis. Section 6 compares the proposed filter with the state of the art. Finally, conclusions are drawn in Section 7.

2. The Proposed Filter Design

Figure 1 depicts the proposed ultra-low-power universal Gm-C filter, capable of operating in various filtering modes. i_in1, i_in2, and i_in3 represent current inputs, while v_in1, v_in2, and v_in3 correspond to voltage inputs. v_OUT denotes the output voltage, and i_OUT represents the output current.

The proposed filter can operate in voltage, current, trans-resistance, and transconductance modes. It is composed of three gm blocks; g_m1, g_m2, and g_m3 in Figure 1, respectively. When a transconductance mode is required, a dedicated transconductance mode block g_mT is added to the basic filter (dashed line in Figure 1). Furthermore, to further investigate the transfer function, Figure 2 reports the signal flow graph (SFG) of the proposed circuit. Defining D(s) a polynomial function as:

(1)$\mathrm{D}\left(\mathrm{s}\right)={\mathrm{s}}^2+{\displaystyle \frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}}\mathrm{s}+{\displaystyle \frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}$

The transfer functions that describe the behavior of the universal multi-mode Gm-C filter in the different operating modes are as follows:

(2)${\mathrm{v}}_{\mathrm{O}\mathrm{U}\mathrm{T}\left(\mathrm{VM}\right)}={\displaystyle \frac{\mathrm{D}(\mathrm{s}){\mathrm{v}}_{\mathrm{in}3}+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{g}}_{\mathrm{m}3}{\mathrm{C}}_2}\mathrm{s}{\mathrm{v}}_{\mathrm{in}2}+\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{g}}_{\mathrm{m}3}}{\mathrm{s}}^2{\mathrm{v}}_{\mathrm{in}1}}{\mathrm{D}\left(\mathrm{s}\right)}}$

(3)${\mathrm{i}}_{\mathrm{O}\mathrm{U}\mathrm{T}\left(\mathrm{TCM}\right)}={\displaystyle \frac{{\mathrm{g}}_{\mathrm{mT}}\left[\mathrm{D}(\mathrm{s}){\mathrm{v}}_{\mathrm{in}3}+\mathrm{s}\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{g}}_{\mathrm{m}3}{\mathrm{C}}_2}{\mathrm{v}}_{\mathrm{in}2}+\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{g}}_{\mathrm{m}3}}{\mathrm{s}}^2{\mathrm{v}}_{\mathrm{in}1}\right]}{\mathrm{D}\left(\mathrm{s}\right)}}$

(4)${\mathrm{i}}_{\mathrm{O}\mathrm{U}\mathrm{T}\left(\mathrm{CM}\right)}={\displaystyle \frac{{\mathrm{s}}^2{\mathrm{i}}_{\mathrm{in}1}+\mathrm{s}\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}{\mathrm{i}}_{\mathrm{in}2}+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}{\mathrm{i}}_{\mathrm{in}3}}{\mathrm{D}\left(\mathrm{s}\right)}}$

(5)${\mathrm{v}}_{\mathrm{O}\mathrm{U}\mathrm{T}\left(\mathrm{TRM}\right)}={\displaystyle \frac{{\mathrm{s}}^2{\mathrm{i}}_{\mathrm{in}1}+\mathrm{s}\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}{\mathrm{i}}_{\mathrm{in}2}+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}{\mathrm{i}}_{\mathrm{in}3}}{{\mathrm{g}}_{\mathrm{m}3}\mathrm{D}\left(\mathrm{s}\right)}}$

2.1. Current and Trans-Resistance Modes

When v_in1 = v_in2 = v_in3 = 0, the filter operates in the current mode, and its filtering responses are obtained as:

Low-Pass (LP): if i_in = i_in3; i_in1 = i_in2 = 0

(6)${\displaystyle \frac{{\mathrm{i}}_{\mathrm{OUT}\left(\mathrm{LP}\right)}}{{\mathrm{i}}_{\mathrm{in}}}}={\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}; {\displaystyle \frac{{\mathrm{v}}_{\mathrm{OUT}\left(\mathrm{LP}\right)}}{{\mathrm{i}}_{\mathrm{in}}}}=-{\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{{\mathrm{g}}_{\mathrm{m}3}\mathrm{D}\left(\mathrm{s}\right)}}$

High-Pass (HP): if i_in = i_in1; i_in2 = i_in3 = 0

(7)${\displaystyle \frac{{\mathrm{i}}_{\mathrm{OUT}\left(\mathrm{HP}\right)}}{{\mathrm{i}}_{\mathrm{in}}}}={\displaystyle \frac{{\mathrm{s}}^2}{\mathrm{D}\left(\mathrm{s}\right)}}; {\displaystyle \frac{{\mathrm{v}}_{\mathrm{OUT}\left(\mathrm{HP}\right)}}{{\mathrm{i}}_{\mathrm{in}}}}=-{\displaystyle \frac{{\mathrm{s}}^2}{{\mathrm{g}}_{\mathrm{m}3}\mathrm{D}\left(\mathrm{s}\right)}}$

Band-Pass (BP): if i_in = i_in2; i_in1 = i_in3 = 0

(8)${\displaystyle \frac{{\mathrm{i}}_{\mathrm{OUT}\left(\mathrm{BP}\right)}}{{\mathrm{i}}_{\mathrm{in}}}}={\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{s}}{\mathrm{D}\left(\mathrm{s}\right)}}; {\displaystyle \frac{{\mathrm{v}}_{\mathrm{OUT}\left(\mathrm{BP}\right)}}{{\mathrm{i}}_{\mathrm{in}}}}=-{\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{s}}{{\mathrm{g}}_{\mathrm{m}3}\mathrm{D}\left(\mathrm{s}\right)}}$

Band-Reject (BR): if i_in = i_in1 = i_in3; i_in2 = 0

(9)${\displaystyle \frac{{\mathrm{i}}_{\mathrm{OUT}\left(\mathrm{BR}\right)}}{{\mathrm{i}}_{\mathrm{in}}}}={\displaystyle \frac{{\mathrm{s}}^2+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}; {\displaystyle \frac{{\mathrm{v}}_{\mathrm{OUT}\left(\mathrm{BR}\right)}}{{\mathrm{i}}_{\mathrm{in}}}}=-{\displaystyle \frac{{\mathrm{s}}^2+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{{\mathrm{g}}_{\mathrm{m}3}\mathrm{D}\left(\mathrm{s}\right)}}$

All-Pass (AP): if i_in = i_in1 = i_in2 = i_in3

(10)${\displaystyle \frac{{\mathrm{i}}_{\mathrm{OUT}\left(\mathrm{AP}\right)}}{{\mathrm{i}}_{\mathrm{in}}}}={\displaystyle \frac{{\mathrm{s}}^2+\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{s}+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}; {\displaystyle \frac{{\mathrm{v}}_{\mathrm{OUT}\left(\mathrm{AP}\right)}}{{\mathrm{i}}_{\mathrm{in}}}}=-{\displaystyle \frac{{\mathrm{s}}^2+\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{s}+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{{\mathrm{g}}_{\mathrm{m}3}\mathrm{D}\left(\mathrm{s}\right)}}$

2.2. Voltage and Transconductance Modes

When i_in1 = i_in2 = i_in3 = 0, the voltage mode filtering responses are obtained as:

Low-Pass (LP): if v_in = −v_in1 = −v_in2 = v_in3

(11)${\displaystyle \frac{{\mathrm{v}}_{\mathrm{OUT}\left(\mathrm{LP}\right)}}{{\mathrm{v}}_{\mathrm{in}}}}={\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}; {\displaystyle \frac{{\mathrm{i}}_{\mathrm{OUT}\left(\mathrm{LP}\right)}}{{\mathrm{v}}_{\mathrm{in}}}}=-{\mathrm{g}}_{\mathrm{mT}}{\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}$

High-Pass (HP): if v_in = v_in1; v_in2 = v_in3 = 0

(12)${\displaystyle \frac{{\mathrm{v}}_{\mathrm{OUT}\left(\mathrm{HP}\right)}}{{\mathrm{v}}_{\mathrm{in}}}}={\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{g}}_{\mathrm{m}3}}{\mathrm{s}}^2}{\mathrm{D}\left(\mathrm{s}\right)}}; {\displaystyle \frac{{\mathrm{i}}_{\mathrm{OUT}\left(\mathrm{HP}\right)}}{{\mathrm{v}}_{\mathrm{in}}}}=-{\mathrm{g}}_{\mathrm{mT}}{\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{g}}_{\mathrm{m}3}}{\mathrm{s}}^2}{\mathrm{D}\left(\mathrm{s}\right)}}$

Band-Pass (BP): if v_in = v_in2; v_in1 = v_in3 = 0

(13)${\displaystyle \frac{{\mathrm{v}}_{\mathrm{OUT}\left(\mathrm{BP}\right)}}{{\mathrm{v}}_{\mathrm{in}}}}={\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{g}}_{\mathrm{m}3}{\mathrm{C}}_2}\mathrm{s}}{\mathrm{D}\left(\mathrm{s}\right)}}; {\displaystyle \frac{{\mathrm{i}}_{\mathrm{OUT}\left(\mathrm{BP}\right)}}{{\mathrm{v}}_{\mathrm{in}}}}=-{\mathrm{g}}_{\mathrm{mT}}{\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{g}}_{\mathrm{m}3}{\mathrm{C}}_2}\mathrm{s}}{\mathrm{D}\left(\mathrm{s}\right)}}$

Band-Reject (BR): if v_in = −v_in2 = v_in3; v_in1 = 0

(14)${\displaystyle \frac{{\mathrm{v}}_{\mathrm{OUT}\left(\mathrm{BR}\right)}}{{\mathrm{v}}_{\mathrm{in}}}}={\displaystyle \frac{{\mathrm{s}}^2+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}; {\displaystyle \frac{{\mathrm{i}}_{\mathrm{OUT}\left(\mathrm{BR}\right)}}{{\mathrm{v}}_{\mathrm{in}}}}=-{\mathrm{g}}_{\mathrm{mT}}{\displaystyle \frac{{\mathrm{s}}^2+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}$

All-Pass (AP): if v_in = v_in3; v_in1 = v_in2 = 0

(15)${\displaystyle \frac{{\mathrm{v}}_{\mathrm{OUT}\left(\mathrm{AP}\right)}}{{\mathrm{v}}_{\mathrm{in}}}}={\displaystyle \frac{{\mathrm{s}}^2+\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{s}+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}; {\displaystyle \frac{{\mathrm{i}}_{\mathrm{OUT}\left(\mathrm{AP}\right)}}{{\mathrm{v}}_{\mathrm{in}}}}=-{\mathrm{g}}_{\mathrm{mT}}{\displaystyle \frac{{\mathrm{s}}^2+\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{s}+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}$

Furthermore, the filter performance parameters such as the center frequency ${\mathsf{\omega }}_0$ and the quality factor Q can be calculated as:

(16)${\mathsf{\omega }}_0=\sqrt{{\displaystyle \frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}}$

(17)$\mathrm{Q}=\sqrt{{\displaystyle \frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{C}}_2}{{\mathrm{g}}_{\mathrm{m}1}{\mathrm{C}}_1}}}$

Table 1 summarizes how different filtering functions come from a different setup of the inputs universal Gm-C filter.

3. Simulation Results

3.1. Proposed OTA and Gm-C Structures

The filter’s performance has been verified using the 180 nm TSMC technology process. The operational transconductance amplifier (OTA), which is the building block of the proposed filter, as well as the g_mT block required for the transconductance mode, are depicted in Figure 3a and Figure 3b, respectively. The body terminals of the NMOS transistors are tied to the ground, while the body terminals of the PMOS are usually tied to the supply voltage V_DD, whether differently specified or not. In fact, the body terminals of the PMOS transistors highlighted in red in Figure 3 are connected together, and available for proper biasing. This voltage allows the center frequency of the Gm-C filter to be adjusted whenever process, supply voltage, and temperature (PVT) variations occur. The inverter-based topology can provide a transconductance gain while the circuit minimizes its power consumption. In particular, the gain of the proposed OTA is:

(18)${\mathrm{A}}_{\mathrm{V}}={\displaystyle \frac{({\mathrm{g}}_{\mathrm{m}18,21}+{\mathrm{g}}_{\mathrm{m}17,22})·({\mathrm{g}}_{\mathrm{m}2,11}+{\mathrm{g}}_{\mathrm{m}3,12}+{\mathrm{g}}_{\mathrm{m}6,9}+{\mathrm{g}}_{\mathrm{m}7,10})}{{\mathrm{g}}_{\mathrm{m}13,14}}}·({\mathrm{r}}_{\mathrm{d}18,21}\parallel {\mathrm{r}}_{\mathrm{d}16,19})$

Table 2 summarizes the transistor aspect ratios, while Table 3 lists the features of the proposed OTA structure (Figure 3) employed in the Gm-C filter, such as DC gain, gain-bandwidth product (GBW), phase margin, CMRR, and PSRR, referring to a capacitance load value (C_L) of 1 pF. Then, the AC simulation results for the gain and phase of the proposed OTA are shown in Figure 4. Notice that two output replicas (shown in gray in Figure 3) provide additional voltages (V_OUT+,2 and V_OUT+,3) as outputs of the transconductance g_m3 only in the proposed filter (see Figure 1). A still-inverter-based Common-mode Feedback (CMFB) circuit is made by the transistor M29-M30 and M31-M32 with a common mode voltage value of V_CM = 0.3 V (see Figure 3). The circuit exhibits an input common mode dynamic range from 0.1 V to 0.4 V.

3.2. Gm-C Structures

Figure 5 illustrates the simulation results for the general proposed multi-mode Gm-C filter, while the transconductance (g_m) and capacitance (C₁ = C₂) values are 58 nS and 20 pF.

3.3. PVT Analysis

A Monte-Carlo analysis is performed to find out how process and mismatch variations affect the center frequency of the proposed Gm-C filter. Figure 6 shows the band-pass frequency response for 1000 iterations. Moreover, a complete PVT variation analysis has been performed. In particular, the center frequency of the Gm-C filter is investigated under different corner processes in Table 4. Table 5 refers to the variation in the supply voltage (−/+10%), while Table 6 considers the temperature variation in a temperature range from 0 °C to 40 °C. From these tables, a significant variation in the center frequency of the proposed filter is shown. This can also be highlighted in Figure 7, Figure 8 and Figure 9. Figure 7 depicts the band-pass frequency responses for the Gm-C filter proposed in various corners. Figure 8 and Figure 9 illustrate the effects of supply voltage variations from 0.45 V to 0.55 V and temperature variations from 0 °C to 40 °C on the band-pass and low-pass filters, respectively.

3.4. Bulk-Biasing Technique

To compensate for the shift of the center frequency of the proposed Gm-C filter due to the PVT variations, calibration by fine-tuning the body bias of the PMOS transistors of the transconductance blocks (see V_CAL terminals in Figure 3) is considered. From this perspective, Table 7 shows the values of V_CAL to compensate for the variation in the center frequency across the five-corner process. Notice that the center frequency is affected in the corner process SS and FS and SF and FF in 132 mV and 100 mV of drift from the supply voltage allows for the re-centering of the filter on the frequency of 462 Hz. Table 8 shows how a variation of +/− 10% on the supply voltage can be compensated with only 2 mV changes of V_CAL. Table 9 reports how the lower temperature affected the center frequency of the filter. Thus, the circuit is suitable for integrated systems for indoor applications.

Figure 10 shows a match between the theoretical (given by a math calculation) and simulation results for the proposed Gm-C filter. In particular, the simulation finds a frequency value of 422 Hz, while the theoretical one is ${\mathrm{f}}_0={\displaystyle \frac{1}{2\mathsf{\pi }}}\sqrt{{\displaystyle \frac{g_{\mathrm{m}2}{g}_{\mathrm{m}1}}{C_2{C}_1}}}=426 \mathrm{Hz}$.

3.5. Group Delay of the Band-Pass Filter

The general transfer function for the band-pass filter is:

(19)$\mathrm{H}(\mathrm{s})={\displaystyle \frac{\left[\frac{{\mathsf{\omega }}_0}{\mathrm{Q}}\right]\mathrm{s}}{{\mathrm{s}}^2+\left[\frac{{\mathsf{\omega }}_0}{\mathrm{Q}}\right]\mathrm{s}+{{\mathsf{\omega }}_0}^2}}$

where ${\mathsf{\omega }}_0$ is the center pulsation and Q is the quality factor of the filter. The group delay for the band-pass filter is:

(20)$\mathrm{D}(\mathsf{\omega })={\displaystyle \frac{\left[\frac{{\mathsf{\omega }}_0}{\mathrm{Q}}\right]\left[{{\mathsf{\omega }}_0}^2+{\mathsf{\omega }}^2\right]}{{\left[{{\mathsf{\omega }}_0}^2-{\mathsf{\omega }}^2\right]}^2+{\left[\frac{{\mathsf{\omega }}_0}{\mathrm{Q}}\right]}^2{\mathsf{\omega }}^2}}$

Thus, the maximum value of the group delay for the band-pass filter is:

(21)$\mathrm{D}(\mathsf{\omega })={\displaystyle \frac{2}{\left[\frac{{\mathsf{\omega }}_0}{\mathrm{Q}}\right]}}={\displaystyle \frac{2{\mathrm{C}}_2}{{\mathrm{g}}_{\mathrm{m}1}}}$

The group delay is shown in Figure 11: 690 µs is the group delay at the filter’s center frequency.

3.6. The Linearity Performance of the Proposed Filter

The proposed filter’s linearity performance is investigated by applying a 40 mVPP sinusoidal input at 10 Hz. Figure 12 shows the input and output transient simulation results for different responses. Furthermore, as the input signal frequency of 10 Hz is outside the pass band of the band-pass and high-pass filters, their output signals are significantly weakened in comparison to those of the filtering responses, which are approximately the same amplitude as the input signals. Also, the input signal frequency of 462 Hz (center frequency) has been applied, as shown in Figure 13. Figure 14 highlights how the total harmonic distortion (THD) of the proposed Gm-C filter varies due to the input voltage amplitude changes between 40 mVpp and 120 mVpp. The THD values for the proposed filter for different signal amplitudes and across the five corner processes are summarized in Table 10 and Table 11, respectively.

4. Noise Analysis

Given the low level of the input signal, the noise performance analysis is critical. To understand qualitatively how design parameters affect the overall input-referred noise, the following equations refer to the saturation region model, assumed as the worst-case scenario. In any case, the subthreshold real values are expected to be lower [28,29,30]. Assuming ${\mathrm{g}}_{\mathrm{mINV}}$, the transconductance at the OTA input terminals is as follows:

(22)${\mathrm{g}}_{\mathrm{mINV}}=\left[{\mathrm{g}}_{\mathrm{m}2}+{\mathrm{g}}_{\mathrm{m}3}+{\mathrm{g}}_{\mathrm{m}9}+{\mathrm{g}}_{\mathrm{m}10}\right]$

The input-referred thermal and flicker noise values for the OTA used in the proposed Gm-C filter are:

(23)$\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{Thermal}}^2}=8\mathrm{KT}\mathsf{\gamma }\left[{\displaystyle \frac{{\mathrm{g}}_{\mathrm{m}14}+{\mathrm{g}}_{\mathrm{mINV}}}{{\mathrm{g}}_{\mathrm{mINV}}^2}}+2{\displaystyle \frac{{{\mathrm{g}}_{\mathrm{m}14}^2(\mathrm{g}}_{\mathrm{m}16}+{\mathrm{g}}_{\mathrm{m}18})}{{\mathrm{g}}_{\mathrm{mINV}}^2 {\mathrm{g}}_{\mathrm{m}18}^2}}\right]$

(24)$$\begin{gathered} \overline{{\mathrm{V}}_{\mathrm{n},\mathrm{Flicker}}^2}={\displaystyle \frac{{2\mathrm{K}}_{\mathrm{P}}}{{\mathrm{C}}_{\mathrm{o}\mathrm{x}}\mathrm{f}}}\left[{\displaystyle \frac{1}{{\left(\mathrm{W}·\mathrm{L}\right)}_2}}+{\displaystyle \frac{1}{{\left(\mathrm{W}·\mathrm{L}\right)}_9}}\right]+{\displaystyle \frac{{2\mathrm{K}}_{\mathrm{N}}}{{\mathrm{C}}_{\mathrm{o}\mathrm{x}}\mathrm{f}}}\left[{\displaystyle \frac{1}{{\left(\mathrm{W}·\mathrm{L}\right)}_3}}+{\displaystyle \frac{1}{{\left(\mathrm{W}·\mathrm{L}\right)}_{10}}}+{\displaystyle \frac{{{\mathrm{g}}_{\mathrm{m}14}}^2}{{{\mathrm{g}}_{\mathrm{mINV}}}^2{\left(\mathrm{W}·\mathrm{L}\right)}_{14}}}\right]+ \\ +{\displaystyle \frac{4}{{\mathrm{C}}_{\mathrm{o}\mathrm{x}}\mathrm{f}}}{\displaystyle \frac{{{\mathrm{g}}_{\mathrm{m}14}}^2}{{{\mathrm{g}}_{\mathrm{mINV}}}^2}}\left\{\left[{\displaystyle \frac{{\mathrm{K}}_{\mathrm{P}}}{{\left(\mathrm{W}·\mathrm{L}\right)}_{16}}}+{\displaystyle \frac{{\mathrm{K}}_{\mathrm{N}}}{{\left(\mathrm{W}·\mathrm{L}\right)}_{18}}}\right]\left(1+{\displaystyle \frac{{{\mathrm{g}}_{\mathrm{m}16}}^2}{{{\mathrm{g}}_{\mathrm{m}18}}^2}}\right)\right\} \end{gathered}$$

where K is the Boltzmann constant, T is the temperature, γ is the noise factor, K_P and K_N are the flicker noise coefficients of the PMOS and NMOS transistors, C_OX is the gate-oxide capacitance, W is the width and L is the length of the transistors. These equations offer design guidelines for noise minimization. The overall noise is:

(25)$\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in},\mathrm{OTA}}^2}=\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{Thermal}}^2}+\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{Flicker}}^2}$

Notice that ${\mathrm{g}}_{\mathrm{mINV}}$ is roughly 4× higher than the transconductance of other transistors in the OTA topology in Figure 3. This results in the minimized input-referred noise of the overall Gm-C filter.

Considering ($\left|{\mathrm{H}}_{\mathrm{N}\mathrm{1,2},3}\left(\mathrm{s}\right)\right|$) as the transfer function for each OTA input-referred noise and ($\left|{\mathrm{H}}_{\mathrm{B}}\left(\mathrm{s}\right)\right|$), the band-pass filter’s transfer function is as follows:

(26)$\left|{\mathrm{H}}_{\mathrm{N}1}\left(\mathrm{s}\right)\right|=\left|{\displaystyle \frac{{\mathrm{V}}_{\mathrm{n},\mathrm{OUT}1}}{{\mathrm{V}}_{\mathrm{n},\mathrm{in}1}}}\right|=\left|{\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{g}}_{\mathrm{m}3}}{\mathrm{S}}^2}{\mathrm{D}\left(\mathrm{s}\right)}}\right|$

(27)$\left|{\mathrm{H}}_{\mathrm{N}2}\left(\mathrm{s}\right)\right|=\left|{\mathrm{H}}_{\mathrm{B}}\left(\mathrm{s}\right)\right|=\left|{\displaystyle \frac{{\mathrm{V}}_{\mathrm{n},\mathrm{OUT}2}}{{\mathrm{V}}_{\mathrm{n},\mathrm{in}2}}}\right|=\left|{\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{g}}_{\mathrm{m}3}{\mathrm{C}}_2}\mathrm{S}}{\mathrm{D}\left(\mathrm{s}\right)}}\right|$

(28)$\left|{\mathrm{H}}_{\mathrm{N}3}\left(\mathrm{s}\right)\right|=\left|{\displaystyle \frac{{\mathrm{V}}_{\mathrm{n},\mathrm{OUT}3}}{{\mathrm{V}}_{\mathrm{n},\mathrm{in}3}}}\right|=\left|{\displaystyle \frac{{\mathrm{s}}^2+\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{s}+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}\right|$

The input-referred noise of the three transconductance blocks for the band-pass filter (V_n,in,in1, V_n,in,in1, V_n,in,in3 in Figure 15) can be expressed by:

(29)$\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in},\mathrm{in}1}^2}={\left|{\displaystyle \frac{{\mathrm{V}}_{\mathrm{n},\mathrm{OUT}1}}{{\mathrm{H}}_{\mathrm{B}}\left(\mathrm{s}\right)}}\right|}^2=\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in}1}^2}{\left|{\displaystyle \frac{{\mathrm{H}}_{\mathrm{N}1}\left(\mathrm{s}\right)}{{\mathrm{H}}_{\mathrm{B}}\left(\mathrm{s}\right)}}\right|}^2=\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in}1}^2}{\left[{\displaystyle \frac{{\mathrm{C}}_2\mathrm{s}}{{\mathrm{g}}_{\mathrm{m}2}}}\right]}^2$

(30)$\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in},\mathrm{in}2}^2}={\left|{\displaystyle \frac{{\mathrm{V}}_{\mathrm{n},\mathrm{OUT}2}}{{\mathrm{H}}_{\mathrm{B}}\left(\mathrm{s}\right)}}\right|}^2=\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in}2}^2}{\left|{\displaystyle \frac{{\mathrm{H}}_{\mathrm{N}2}\left(\mathrm{s}\right)}{{\mathrm{H}}_{\mathrm{B}}\left(\mathrm{s}\right)}}\right|}^2=\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in}2}^2}$

(31)$\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in},\mathrm{in}3}^2}={\left|{\displaystyle \frac{{\mathrm{V}}_{\mathrm{n},\mathrm{OUT}3}}{{\mathrm{H}}_{\mathrm{B}}\left(\mathrm{s}\right)}}\right|}^2=\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in}3}^2}{\left|{\displaystyle \frac{{\mathrm{H}}_{\mathrm{N}3}\left(\mathrm{s}\right)}{{\mathrm{H}}_{\mathrm{B}}\left(\mathrm{s}\right)}}\right|}^2=\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in}3}^2}{\left[{\displaystyle \frac{{\mathrm{g}}_{\mathrm{m}3}{\mathrm{C}}_2\mathrm{D}\left(\mathrm{s}\right)}{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}\mathrm{s}}}\right]}^2$

Thus, the overall equivalent input-referred noise for the band-pass filter $\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in},\mathrm{BP}}^2}$ is:

(32)$\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in},\mathrm{BP}}^2}=\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in},\mathrm{in}1}^2}+\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in},\mathrm{in}2}^2}+\overline{{\mathrm{V}}_{\mathrm{n},\mathrm{in},\mathrm{in}3}^2}$

5. Sensitivity Analysis

Naming Š the sensitivity of the K circuit characteristic with respect to the L parameter, $Š$ is defined as:

(33)${\mathrm{Š}}_{\mathrm{L}}^{\mathrm{K}}={\displaystyle \frac{\partial \mathrm{K}}{\partial \mathrm{L}}}·{\displaystyle \frac{\mathrm{L}}{\mathrm{K}}}$

For instance, the sensitivity of the g_m2 for the current-mode low-pass filter is calculated in (34), which is similar to the sensitivity analysis for g_m2 in (35).

(34)${\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{L}\mathrm{P}\mathrm{i}}={\displaystyle \frac{\partial \mathrm{L}{\mathrm{P}}_{\mathrm{i}}}{{\partial \mathrm{g}}_{\mathrm{m}2}}}·{\displaystyle \frac{{\mathrm{g}}_{\mathrm{m}2}}{\mathrm{L}\mathrm{P}}}={\displaystyle \frac{{\mathrm{g}}_{\mathrm{m}1}{\mathrm{C}}_2{\mathrm{C}}_1\mathrm{D}\left(\mathrm{s}\right)-{\mathrm{g}}_{\mathrm{m}2}{{\mathrm{g}}_{\mathrm{m}1}}^2}{{\mathrm{D}\left(\mathrm{s}\right)}^2}}·{\displaystyle \frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{C}}_2{\mathrm{C}}_1\mathrm{D}\left(\mathrm{s}\right)}{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}}={\displaystyle \frac{{\mathrm{S}}^2+\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{S}}{{\mathrm{D}}_{\left(\mathrm{s}\right)}}}$

5.1. The Sensitivity Analysis of the Current-Mode Filter

The sensitivity of the universal filter responses in the current mode to the capacitance and transconductance values are as follows:

(35)${\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}=-{\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}={\displaystyle \frac{{\mathrm{s}}^2+\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{s}}{\mathrm{D}\left(\mathrm{s}\right)}}$

(36)${\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{B}{\mathrm{P}}_{\mathrm{i}}}=-{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{B}{\mathrm{P}}_{\mathrm{i}}}={\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}$

(37)${\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}=-{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}={\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{B}{\mathrm{P}}_{\mathrm{i}}}=-{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{B}{\mathrm{P}}_{\mathrm{i}}}={\displaystyle \frac{{\mathrm{s}}^2}{\mathrm{D}\left(\mathrm{s}\right)}}$

(38)${\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}=-{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}={\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}$

(39)${\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}=-{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}={\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{s}+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}$

(40)${\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{B}{\mathrm{P}}_{\mathrm{i}}}={\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{\mathrm{C}1}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}$

(41)${\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{B}{\mathrm{R}}_{\mathrm{i}}}={\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}$

(42)${\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_1}^{\mathrm{B}{\mathrm{R}}_{\mathrm{i}}}={\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_1}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_1}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}$

(43)${\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_2}^{\mathrm{B}{\mathrm{R}}_{\mathrm{i}}}={\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_2}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_2}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}$

Notice that the sum of the sensitivity values to the all-filtering responses in the current mode is zero, as reported in the following:

(44)$\begin{array}{c}{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{L}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{B}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{B}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{B}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{B}{\mathrm{P}}_{\mathrm{i}}}+ \\ +{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{H}{\mathrm{P}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{B}{\mathrm{R}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{B}{\mathrm{R}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{B}{\mathrm{R}}_{\mathrm{i}}}+{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{B}{\mathrm{R}}_{\mathrm{i}}}=0\end{array}$

5.2. The Sensitivity Analysis in Voltage-Mode Filter

The sensitivity of the universal filter responses in the voltage mode to the capacitance and transconductance values are as follows:

(45)${\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}=-{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}={\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{B}{\mathrm{P}}_{\mathrm{v}}}=-{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{B}{\mathrm{P}}_{\mathrm{v}}}={\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}={\displaystyle \frac{{\mathrm{s}}^2}{\mathrm{D}\left(\mathrm{s}\right)}}$

(46)${\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}=-{\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}={\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{B}{\mathrm{P}}_{\mathrm{v}}}={\displaystyle \frac{{\mathrm{s}}^2+\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{s}}{\mathrm{D}\left(\mathrm{s}\right)}}$

(47)${\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{B}{\mathrm{P}}_{\mathrm{v}}}={\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}=-{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}{\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}$

(48)${\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}={\displaystyle \frac{\frac{{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2}\mathrm{s}+\frac{{\mathrm{g}}_{\mathrm{m}2}{\mathrm{g}}_{\mathrm{m}1}}{{\mathrm{C}}_2{\mathrm{C}}_1}}{\mathrm{D}\left(\mathrm{s}\right)}}$

(49)$-{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}3}}^{\mathrm{B}{\mathrm{P}}_{\mathrm{v}}}=-{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}3}}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}=1$

(50)${\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{B}{\mathrm{R}}_{\mathrm{v}}}={\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{\mathrm{C}1}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}$

(51)${\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{B}{\mathrm{R}}_{\mathrm{v}}}={\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}$

(52)${\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_1}^{\mathrm{B}{\mathrm{R}}_{\mathrm{v}}}={\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_1}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_1}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}$

(53)${\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_2}^{\mathrm{B}{\mathrm{R}}_{\mathrm{v}}}={\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_2}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{\mathrm{g}{\mathrm{m}}_2}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}$

Again, the sum of the sensitivity values to the all-filtering responses, also in the voltage mode, is zero:

(54)$\begin{array}{c}{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{L}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{B}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{B}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{B}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{B}{\mathrm{P}}_{\mathrm{v}}}+ \hfill \\ {\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}1}}^{\mathrm{B}{\mathrm{R}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{C}}_1}^{\mathrm{B}{\mathrm{R}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}2}}^{\mathrm{B}{\mathrm{R}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{C}}_2}^{\mathrm{B}{\mathrm{R}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}3}}^{\mathrm{B}{\mathrm{P}}_{\mathrm{v}}}+{\mathrm{Š}}_{{\mathrm{g}}_{\mathrm{m}3}}^{\mathrm{H}{\mathrm{P}}_{\mathrm{v}}}=0\hfill \end{array}$

6. Comparison with the State of the Art

Table 12 compares the proposed Gm-C circuit with the state of the art. The proposed filter shows a lower rms input-referred noise than [31,32,33,34,35,36]. Additionally, the proposed circuit consumes less power and even the figure-of-merit (FOM) for the filter. It is defined as:

(55)$\mathrm{FOM}={\displaystyle \frac{\mathrm{P}}{\mathrm{f}\cdot \mathrm{N}\cdot \mathrm{DR}}}$

where P is the power consumption, f is the center frequency of the Gm-C filter, N is its order, and DR is the dynamic range.

7. Conclusions

An ultra-low-power, low-voltage Gm-C filter capable of producing various filtering responses (LP, HP, AP, BP, BR) in four-mode filtering operations has been designed in a 180 nm TSMC technology node. The Gm-C filter performance at a center frequency of 462 Hz has been shown in this paper. Body-bias-driven compensations for all the frequency responses under the PVT variations have also been reported. Also, the THD, the overall input-referred noise, and the sensitivity have been considered. The proposed filter operates at 0.5 V supply voltage with the minimum number of gm blocks, with its building transistors operating in the subthreshold region, showing an overall power consumption of 32 nW.

Author Contributions

Conceptualization, A.N.; methodology, A.N.; validation, A.N.; formal analysis, A.N.; investigation, A.N., O.A, and D.D.C.; resources, O.A. and D.D.C.; data curation, A.N., O.A., and D.D.C.; writing—original draft preparation, A.N.; writing—review and editing, O.A. and D.D.C.; visualization, A.N.; supervision, O.A. and D.D.C.; project administration, O.A., and D.D.C.; funding acquisition, O.A., and D.D.C. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data is contained within the article.

Acknowledgments

The authors would like to thank Europractice and TSMC for PDK access.

Conflicts of Interest

The authors declare no conflict of interest.

Abbreviations

TSMC	Taiwan Semiconductor Manufacturing Company
OTA	Transconductance Operational Amplifiers
VM	Voltage Mode
CM	Current Mode
TCM	Transconductance-Mode
TRM	Trans-resistance Mode
LP	Low-Pass
HP	High-Pass
BP	Band-Pass
BR	Band-Reject
AP	All-Pass

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. The proposed universal multi-mode Gm-C filter.

Figure 2. The signal flow graph of the Gm-C proposed filter.

Figure 3. The circuits used in the proposed Gm-C filter (a) The proposed OTA (gray branches refer to gm3 block only and in red, the terminal for the calibration. (b) Transconductance mode gmT block.

Figure 4. The OTA AC responses: (a) gain response; (b) phase response.

Figure 5. The frequency responses of the proposed Gm-C filter in the various modes: (a) voltage mode; (b) transconductance mode; (c) current mode; (d) trans-resistance mode.

Figure 6. Monte-Carlo simulation results for the center frequency of the band-pass filter.

Figure 7. Variations in corner technology for the band-pass frequency response.

Figure 8. Variations in supply voltage for the band-pass frequency response.

Figure 9. Variations in temperature for the low-pass frequency response.

Figure 10. Comparison between simulation and theoretical results for the proposed Gm-C filter.

Figure 11. Group delay for the proposed band-pass filter.

Figure 12. Transient simulation results for the proposed filter: (a) input (10 Hz); (b) output.

Figure 13. Transient simulation results for the proposed band-pass and high-pass filters at the center frequency (462 Hz): (a) input; (b) output.

Figure 14. THD versus voltage input amplitudes.

Figure 15. Modeling of the noise equivalent circuit for the proposed universal filter.

Table 1

The filtering functions of the proposed universal multi-mode Gm-C filter.

Filtering Function	Input for Current andTrans-Resistance Modes	Input for Voltage andTransconductance Modes
LP	iin3	−vin1 = −vin2 = vin3
HP	iin1	vin1
BP	iin2	vin2
BR	iin1 = iin3	−vin2 = vin3
AP	iin1 = iin2 = iin3	vin3

Table 2

The aspect ratio of the OTA transistors employed in the proposed filter.

Aspect Ratio of OTA
Transistor	W/L [µm/µm]
M1, M3, M5, M7, M10, M12	1/0.3 = 3.33
M2, M4, M6, M8, M9, M11	4/0.3 = 13.33
M13, M14	3/0.3 = 10
M15–M22	3/1 = 3
M29–M32	1/0.18 = 5.56
Aspect ratio of transconductance mode gmT block
Transistor	W/L [µm/µm]
M23, M25, M28	1/0.18 = 5.56
M24, M26, M27	4/0.18 = 22.22

Table 3

Characteristics of the proposed OTA used in the proposed filter.

Specification	Value
Supply voltage	0.5V
DC gain	46.6 dB
Phase margin	86°
GBW	17.5 kHz
CMRR	48 dB
PSRR	44 dB
Input-referred noise	$503\mathrm{n}{\displaystyle \frac{\mathrm{V}}{\sqrt{\mathrm{H}\mathrm{z}}}}$
Power consumption	6.3 nW
CL	1 pF

Table 4

The corner variations of the proposed Gm-C design.

	SS	SF	TT	FS	FF
Power consumption	8.3 nW	94.5 nW	32 nW	10 nW	118 nW
Center frequency	132 Hz	1.34 kHz	462 Hz	129 Hz	1.43 kHz

Table 5

The supply voltage variations (−/+10%) of the proposed Gm-C design.

	0.45 V	0.5 V	0.55 V
Power consumption	42 nW	32 nW	32.7 nW
Center frequency	693.4 Hz	462 Hz	305.5 Hz

Table 6

The temperature variations of the proposed Gm-C design.

	0 °C	10 °C	27 °C	40 °C
Power consumption	10 nW	16 nW	32 nW	41.5 nW
Center frequency	164.5 Hz	247 Hz	462 Hz	760 Hz

Table 7

Proposed filter’s center frequency regulation using the body-bias tuning V_CAL across the corner process.

	SS	SF	TT	FS	FF
Body bias of M2, M4, M6, M8, M9, M11	0.368 V	0. 4 V	0.5 V	0.368 V	0.4 V
Power consumption	28.4 nW	31 nW	32 nW	32.2 nW	38 nW
Center frequency	462 Hz	462 Hz	462 Hz	462 Hz	462 Hz

Table 8

Proposed filter’s center frequency regulation using the body-bias tuning V_CAL for supply voltage changing +/− 10%.

	0.45 V	0.5 V	0.55 V
Body bias of M2, M4, M6, M8, M9, M11	0.498 V	0.5 V	0.498 V
Power consumption	27.8 nW	32 nW	43.4 nW
Center frequency	462 Hz	462 Hz	462 Hz

Table 9

Proposed filter’s center frequency regulation using the body-bias tuning V_CAL at different temperatures.

	0 °C	10 °C	27 °C	40 °C
Body bias of M2, M4, M6, M8, M9, M11	0.39 V	0.43 V	0.5 V	0.45 V
Power consumption	28 nW	29.3 nW	32 nW	33 nW
Center frequency	462 Hz	462 Hz	462 Hz	462 Hz

Table 10

THD performance of the proposed filter for different signal amplitudes.

Input VoltageAmplitude (mVPP)	THD% at 10 Hz
	LP	BP	BR	AP	HP
40	0.68	1	0.7	0.3	1.7
60	1	1.6	1.08	0.5	2.4
80	1.55	2.5	1.55	0.7	3.4
100	2	3.6	2	1	5
120	3	5	2.9	1.5	7

Table 11

THD performance of the proposed filter for various corner parameters.

Corner Process	THD% at 10 Hz
	LP	BP	BR	AP	HP
SS	2.3	3.7	2.4	0.2	2.8
SF	4.8	6.6	4.8	4.7	8
FS	2	3.3	2.2	0.17	2
FF	0.5	0.3	0.48	0.47	0.94
TT	0.68	1	0.7	0.3	1.7

Table 12

Gm-C filter’s state-of-the-art comparison referring to their band-pass’ central frequency.

	[31]	[32]	[33]	[34]	[35]	[36]	[37]	This Work
Supply voltage [V]	0.6	0.5	0.5	0.5	0.5	0.5	0.5	0.5
Universal	Yes	Yes	Yes	Yes	Yes	Yes	Yes	Yes
Multi-mode	Yes	Voltage	Yes	Yes	Yes	Voltage	Voltage	Yes
Filter order	2	2	2	2	2	2	2	2
Center frequency [Hz]	5000	254	211	323	114	153	10	462
Dynamic range [dB]	53.2	49.7	58.23	53.2	53.2	50	63	43.59
Rms input-refer. noise [µVrms]	155	116	130	108	208	220	45	93.5
Power consumption [µW]	5.77	0.616	0.281	0.646	0.058	0.037	0.053	0.032
$\mathrm{FOM}[{10}^{-12}$ $\mathrm{W}·{\mathrm{H}\mathrm{z}}^{-1}·{\mathrm{d}\mathrm{B}}^{-1}$]	1.26	3.96	0.816	2.187	0.556	2.41	1.88	0.229

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