Systematic Realization of VDGA-Based Comb Filter for Biomedical Signal Processing

Healthcare professionals rely on biomedical equipment to diagnose and treat patients, recording critical biomedical signals such as electrocardiograms (ECG) [1], electroencephalograms (EEG) [2], and electromyograms (EMG) [3]. These signals are physical and physiological indicators of body functions in real-time. Biomedical signals play a vital role in diagnosing and monitoring health conditions, providing valuable information about one's physiological, pathophysiological, and emotional states. These signals typically contain significant low-frequency components, ranging from several millihertz to several hundreds of hertz, with low amplitudes, often in the range of tens of microvolts to several millivolts.

For example, the ECG signal exhibits amplitudes between 100$\mu \mathrm{V}$ and 5mV, with varying frequencies in the range of 10-200Hz [1]. Similarly, the basic brain signal consists of four brain wave bands-$\delta$ (1-4Hz), $\theta$ (4-8Hz), $\alpha$ (8-13Hz), and $\beta$ (13-40Hz)-constituting the EEG signal. These waves manifest as continuous oscillating curves with weak amplitudes ranging from 2 to 200$\mu \mathrm{V}$ [2].

However, these signals are often susceptible to various types of noise, including instrumentation noise, muscle contractions, electrosurgical interference, baseline drift, motion artifacts, electrode contact issues, and Power-Line Interference (PLI) [4]. These noise sources can compromise the authenticity of biomedical signals, potentially leading to misinterpretations and false alarms. Notably, PLI at 50Hz or 60Hz and its harmonics are the most common noise sources that significantly impact biomedical signals [5].

To address the noise issues in biomedical signals, two commonly employed methods are: 1) A strategy based on the Signal Quality Index (SQI) to assess the clinical acceptability of recorded biomedical signals and 2) An approach based on denoising to reduce noise in biomedical signal recordings [6-11].

Signal Quality Index (SQI) Method: Before feature extraction, the quality of biomedical signals is assessed using various signal quality evaluation techniques. An extensive collection of SQIs is computed from the raw biomedical signal and physiological characteristics. These SQIs are then integrated or used as weighting variables to reduce false alarms in most biomedical equipment.

However, it's important to note that the effectiveness of SQI methods depends on factors such as the dataset, measurement parameters, and threshold selection logic. Additionally, the accuracy of signal quality predictions can be affected by labeling techniques, and the diversity of datasets also plays a significant role.

A method based on denoising: The elimination of power-line interference (PLI) has been addressed by several authors using both digital and analog filters. Many studies have documented various techniques employing different kinds of analog filters to eliminate or attenuate the 50Hz/60Hz signal along with its odd harmonics since analog filters are better suited for real-time processing [12-33]. Such analog filters, designed to notch out multiple frequencies simultaneously, are called comb filters due to the comb shape of the frequency response.

An active comb filter, based on operational amplifiers (op-amps), is introduced in the research [12] to eliminate 60Hz power line interference (PLI), while a similar approach is applied in the research [13] to address 50Hz PLI. Authors [14] also employ op-amps as an active block to mitigate 60Hz PLI and its three odd harmonics at 180Hz, 300Hz, and 420Hz. A notch filter utilizing the second-generation current conveyor (CCII) in 0.35$\mu \mathrm{m}$ CMOS technology is designed to notch out 50Hz interference [15].

Further, Ranjan et al. [16] present an operational transconductance amplifier (OTA) and capacitor-based comb filter circuit, implemented in 0.5$\mu \mathrm{m}$ CMOS technology, tested against 60Hz PLI and its three odd harmonics. The work [17] introduces a comb filter circuit using $\text { CCII+ }$ and IC AD844 in 0.18$\mu \mathrm{m}$ CMOS technology to eliminate undesirable 60Hz PLI and its three odd harmonics at 180Hz, 300Hz, and 420Hz.

In the domain of EEG applications, Qian et al. [18] propose an OTA-C-based low-pass notch filter in 0.35$\mu \mathrm{m}$ CMOS technology. Additionally, for cardiac activity detection, Lee and Cheng [19] introduce another OTA-C filter using 0.18$\mu \mathrm{m}$ CMOS technology. Moreover, Alzaher et al. [20] suggest an op-amp-based power-line removal filter in CMOS 0.18$\mu \mathrm{m}$ for biomedical potential acquisition systems.

To tackle 50Hz PLI and its three odd harmonics (150Hz, 250Hz, and 350Hz), Paul et al. [21] propose a $\text { CCII } \pm$ based filter, where CCIIs are designed using both the macro-model of IC AD844 and dynamic threshold-voltage metal oxide semiconductor (DTMOS) technology. A digitally programmable OTA-based filter is presented in the research [22] to minimize the impact of 50Hz PLI and its 3rd harmonics (150Hz).

For processing biomedical signals at front-ends, Machha Krishna and Laxminidhi [23] introduce a Butterworth transconductor-capacitor (gm-C) filter based on OTA. A multiple-output fully differential OTA (MOFD OTA)-based filter is designed to process electromyography (EMG) signals [24].

Notch filters to eliminate 60Hz frequency are proposed in the studies [25, 26] using op-amps, with the latter employing 65nm CMOS technology. Employing 90nm CMOS technology, the authors [27] introduce an OTA-C filter structure for notch and lowpass filtering to suppress the 50Hz frequency. Furthermore, Zhang et al. [28] propose another OTA-based low-pass filter and a notch filter in 0.18$\mu \mathrm{m}$ CMOS technology.

A multiple-output OTA (MO-OTA)-based notch filter in 0.25$\mu \mathrm{m}$ CMOS technology is suggested to remove PLI [29]. Voltage-mode biquad filters based on voltage differencing differential input buffered amplifiers (VD-DIBA) are proposed for biosensor applications [30]. Similarly, Kumngern et al. [31] suggest using a low-voltage differential difference transconductance amplifier (DDTA) to remove PLI for biosensor applications.

For advanced configurations, Kumngern et al. [32] implement a multiple-input, multiple-output (MIMO) OTA-based notch filter using 0.18$\mu \mathrm{m}$ DTMOS technology. Finally, a multiple-input differential difference transconductance amplifier (MI-DDTA)-based notch filter is proposed in 0.18$\mu \mathrm{m}$ CMOS technology [33].

The existing comb filters present several noteworthy limitations. Firstly, the active comb filter is confined to the removal of a single frequency, i.e., only the fundamental frequency of the PLI [12, 13, 15, 18, 19, 23-30, 33]. Furthermore, using excessive active blocks and passive components in the circuit design, as observed in studies [12-33], results in complex circuits, increasing the chip area and power consumption. Moreover, the design's dependency on resistors contributes to a larger chip area [12-15, 17, 20, 21, 25, 28, 30-31, 33]. Another critical drawback is the absence of an examination and mitigation strategy for the non-idealities inherent in the active block [13, 15-18, 21-27, 29]. Lastly, the structures outlined in studies [12-33] exhibit elevated levels of total harmonic distortion (THD), emphasizing that lower THD values are indicative of superior filter performance.

This literature survey provides an in-depth exploration of comb filters employed in signal-processing applications. Specifically, various comb filter designs and implementations are investigated for their efficacy in addressing power line interference (PLI) issues. The surveyed literature showcases a diverse range of comb filter configurations, emphasizing their application in mitigating interference at 60Hz and 50Hz, along with their respective harmonics.

Due to the intrinsic qualities of current-mode active blocks, such as simplified design, better precision, stronger linearity, wider bandwidth, higher slew rate, greater dynamic range, and the capacity to operate at low voltage, they have drawn much interest from academics in recent years. One of the most adaptable and practical current-mode building blocks is the VDGA (voltage differencing gain amplifier) [34-37].

This paper introduces a novel analog comb filter, leveraging a notch filter approach to mitigate or reduce Power Line Interference (PLI) in low-frequency biomedical signals, including ECG, EEG, and EMG signals. Notably, the proposed circuit exhibits versatility, extending its applicability to diverse scenarios. Beyond its primary use in minimizing PLI in biomedical signals, the circuit can effectively contribute to tasks such as mitigating acoustic feedback oscillations in voice amplification systems and estimating pitch in audio signals within noisy environments. The core design of the filter exclusively features Voltage Differencing Gain Amplifiers (VDGAs) and capacitors, establishing it as a VDGA-based comb filter suitable for IC fabrication.

The significant novelties of the proposed design can be succinctly outlined as follows:

(a) Design of a simple comb filter exclusively comprised of VDGAs and capacitors.

(b) The proposed filter structure occupies less chip area as it does not require any resistor for its design.

(c) Achieving single-frequency notching with remarkable efficiency, utilizing only a single VDGA and two capacitors.

(d) For the blocking of n frequencies, the proposed design employs n VDGAs and 2n capacitors exclusively.

(e) The proposed filter effectively suppresses the effect of the 50Hz PLI along with odd harmonics of the 50Hz PLI, such as 150Hz, 250Hz, and 350Hz.

(f) The filter's parameters, such as the pole frequencies and the quality factors, can be electronically controlled and independently tuned using the VDGA's bias currents.

(h) The proposed structure offers a -42.9dB notch depth sufficient to eliminate the undesired frequency.

(i) The proposed structure achieved the lowest THD of -83dB compared with available structures.

(j) The proposed notch filter is well suited for analog front-end devices for biomedical signal detection, such as EEG, EMG, and ECG.

This research paper is organized into seven distinct sections. Section 2 provides a concise overview of the modern active block, VDGA, which constitutes the fundamental building block in the proposed comb filter. Section 3 delves into the realization of the proposed comb filter by utilizing VDGA. The workability of the proposed design is rigorously assessed and validated in Section 4, employing both the 0.18$\mu \mathrm{m}$ CMOS technology process parameters of TSMC and the macro-model of the IC MAX435. Non-ideality and sensitivity analysis of the proposed comb filter are thoroughly expounded upon in Section 5. Section 6 compares the performance of the proposed comb filter with existing circuits documented in the literature. Finally, Section 7 concludes this paper, encapsulating the key findings and insights derived from the study.

Various analog approaches for designing a comb filter have been reported in the literatures [16-17, 21]. The comb filter is constructed using low Q (quality factor) RLC notch filters, incorporating OTA-based synthetic inductors [16]. Whereas, the methodologies for comb filter design are based on IBPFs (inverted bandpass filters) and APFs (all-pass filters), respectively [17, 21]. Upon scrutinizing these approaches, it becomes evident that the most straightforward and effective means of implementing a comb filter involves using high-Q RLC notch filters. Elevating the quality factor of a band reject/stop filter results in a notch filter capable of selectively attenuating or suppressing specific frequencies within a frequency spectrum.

The systematic realization starts with the analysis of the second-order passive notch filters. Circuit implementations of a series-LC and a parallel-LC notch filter are depicted in Figure 5. In the second-order passive series-LC notch filter, depicted in Figure 5(a), the input signal is fed to the resistor (R), and the output is obtained from a series arrangement of an inductor (L) and a capacitor (C). Conversely, in the second-order passive parallel-LC notch filter, the input is supplied to a parallel arrangement of an inductor (L) and a capacitor (C), and the output is derived across a resistor (R), as illustrated in Figure 5(b).

The standard analysis yields the transfer function of the series-LC notch filter in Figure 5(a) as:

$H\left( s \right)=\frac{{{V}_{out~}}\left( s \right)}{{{V}_{in}}\left( s \right)}=\frac{{{s}^{2}}+{}^{1}/{}_{LC}}{{{s}^{2}}+{}^{sR}/{}_{L}+{}^{1}/{}_{LC}}$      (4)

The parameters for the notch filter are defined as follows:

${{\omega }_{0}}=\frac{1}{\sqrt{LC}}$,${{f}_{0}}=\frac{1}{2\pi \sqrt{LC}}$, and $Q=\frac{1}{R}\sqrt{\frac{L}{C}}$         (5)

Examining the expression for Q in (5), it becomes evident that the Q value is consistently low for practical passive component values. Therefore, this type of filter is suitable for low-Q applications. An effective notch filter demands a narrow, deep stop band around the pole frequency. A High-Q Notch Filter (HQNF) is crucial to achieve these desired properties. Consequently, designing a notch filter with a high Q value is preferable. Thus, the inspiration for implementing an analog comb filter arose, utilizing an HQNF as the foundational circuit.

The transfer function of the series-LC notch filter illustrated in Figure 5(b) is derived through routine analysis:

$H\left( s \right)=\frac{{{V}_{out~}}\left( s \right)}{{{V}_{in}}\left( s \right)}=\frac{{{s}^{2}}+{}^{1}/{}_{LC}}{{{s}^{2}}+{}^{s}/{}_{RC}+{}^{1}/{}_{LC}}$        (6)

From (6), the filter parameters are determined as:

${{\omega }_{0}}=\frac{1}{\sqrt{LC}}$, ${{f}_{0}}=\frac{1}{2\pi \sqrt{LC}}~$, and $Q=R\sqrt{\frac{C}{L}}$           (7)

Observing the expression for Q in Eq. (7), it becomes evident that a high Q value can be achieved with practical passive component values. With this insight, we proceed to design a comb filter, incorporating this fundamental filter in the background. Utilizing the transfer function provided in Eq. (6), the locations of poles and zeros can be plotted in the s-domain, as illustrated in Figure 6.

These poles and zeroes can be expressed as:

${{P}_{1}}=-\frac{1}{2RC}+\frac{1}{2}\sqrt{{{\left( \frac{1}{RC} \right)}^{2}}-\frac{4}{LC}};~{{P}_{2}}=-\frac{1}{2RC}-\frac{1}{2}\sqrt{{{\left( \frac{1}{RC} \right)}^{2}}-\frac{4}{LC}}$; ${{Z}_{1}}=+\frac{j}{\sqrt{LC}}$; ${{Z}_{2}}=-\frac{j}{\sqrt{LC}}$         (8)

For an HQNF, the three essential requirements that the transfer function should exhibit are: (i) poles should be close to zeroes, (ii) a phase shift of -π/2 or π/2 should occur at the pole frequency, and (iii) zeros should be at the imaginary axis. The radius (r) and the angle (θ) in Figure 6 can be expressed as:

$r={{\omega }_{0}}$;    θ =${{\tan }^{-1}}\sqrt{4{{Q}^{2}}-1}$             (9)

Observing (9), it is apparent that θ increases with an increase in the value of Q, indicating that the poles will move toward the zeros with an improvement in Q.

Substituting s= j$\omega $ in (6), the expression for phase can be obtained. The phase of H $(j\omega $) is expressed as:

$\angle H\left( j\omega  \right)=-{{\tan }^{-1}}\frac{\omega {{\omega }_{0}}}{{{Q}_{0}}\left( {{\omega }_{0}}^{2}-{{\omega }^{2}} \right)}$          (10)

At the pole frequency (ω=ω₀), the phase is $\angle H\left( j\omega  \right)$=-π/2, fulfilling the second criterion of the notch filter. Additionally, the zeroes are located on the imaginary axis of the plot.

A cascading methodology is adopted to notch the n-number of frequencies, where the n-number of HQNFs are cascaded. To validate this methodology theoretically, two HQNFs are cascaded, as depicted in Figure 7, for notching two different frequencies. The transfer function of the two cascaded notch filters can be written as:

$H\left( s \right)=\frac{{{V}_{out~}}\left( s \right)}{{{V}_{in}}\left( s \right)}=\left( \frac{{{s}^{2}}+{}^{1}/{}_{{{L}_{1}}{{C}_{1}}}}{{{s}^{2}}+{}^{s}/{}_{{{R}_{1}}{{C}_{1}}}+{}^{1}/{}_{{{L}_{1}}{{C}_{1}}}} \right)\left( \frac{{{s}^{2}}+{}^{1}/{}_{{{L}_{2}}{{C}_{2}}}}{{{s}^{2}}+{}^{s}/{}_{{{R}_{2}}{{C}_{2}}}+{}^{1}/{}_{{{L}_{2}}{{C}_{2}}}} \right)$             (11)

Eq. (11) can be rewritten as:

$H\left( s \right)=\frac{{{V}_{out~}}\left( s \right)}{{{V}_{in}}\left( s \right)}=\frac{{{s}^{4}}+\left( {}^{1}/{}_{{{L}_{1}}{{C}_{1}}}+{}^{1}/{}_{{{L}_{2}}{{C}_{2}}}+{}^{1}/{}_{{{R}_{1}}{{C}_{1}}{{R}_{2}}{{C}_{2}}} \right){{s}^{2}}+{}^{1}/{}_{{{L}_{1}}{{C}_{1}}{{L}_{2}}{{C}_{2}}}}{\left( {{s}^{2}}+{}^{s}/{}_{{{R}_{1}}{{C}_{1}}}+{}^{1}/{}_{{{L}_{1}}{{C}_{1}}} \right)\left( {{s}^{2}}+{}^{s}/{}_{{{R}_{2}}{{C}_{2}}}+{}^{1}/{}_{{{L}_{2}}{{C}_{2}}} \right)}$             (12)

From (12), it can be observed that for larger values of Q₁ and Q₂, ${}^{1}/{}_{{{R}_{1}}{{C}_{1}}{{R}_{2}}{{C}_{2}}\ll }\left( {}^{1}/{}_{{{L}_{1}}{{C}_{1}}}+{}^{1}/{}_{{{L}_{2}}{{C}_{2}}} \right),~$then the transfer function can be expressed as:

$H\left( s \right)=\frac{{{V}_{out~}}\left( s \right)}{{{V}_{in}}\left( s \right)}=\frac{{{s}^{4}}+\left( {}^{1}/{}_{{{L}_{1}}{{C}_{1}}}+~{}^{1}/{}_{{{L}_{2}}{{C}_{2}}} \right){{s}^{2}}+{}^{1}/{}_{{{L}_{1}}{{C}_{1}}{{L}_{2}}{{C}_{2}}}}{\left( {{s}^{2}}+{}^{s}/{}_{{{R}_{1}}{{C}_{1}}}+{}^{1}/{}_{{{L}_{1}}{{C}_{1}}} \right)\left( {{s}^{2}}+{}^{s}/{}_{{{R}_{2}}{{C}_{2}}}+{}^{1}/{}_{{{L}_{2}}{{C}_{2}}} \right)}$              (13)

5.png

Figure 5. Circuits of (a) Series-LC notch filter (b) Parallel LC notch filter

6.png

Figure 6. Pole-zero plot of Figure 5(b)

7.png

Figure 7. Realization of comb filter for n=2

8.png

Figure 8. Pole-zero plot of Figure 7

From (13), the poles (P₁, P₂, P₃, and P₄) and zeroes (Z₁, Z₂, Z₃, and Z₄) can be expressed as:

${{P}_{1}}=-\frac{1}{2~{{R}_{1}}{{C}_{1}}}+\frac{1}{2}\sqrt{{{\left( \frac{1}{{{R}_{1}}{{C}_{1}}} \right)}^{2}}-\frac{4}{{{L}_{1}}{{C}_{1}}}}$; ${{P}_{2}}=-\frac{1}{2~{{R}_{1}}{{C}_{1}}}-\frac{1}{2}\sqrt{{{\left( \frac{1}{{{R}_{1}}{{C}_{1}}} \right)}^{2}}-\frac{4}{{{L}_{1}}{{C}_{1}}}}$; ${{P}_{3}}=-\frac{1}{2~{{R}_{2}}{{C}_{2}}}+\frac{1}{2}\sqrt{{{\left( \frac{1}{{{R}_{2}}{{C}_{2}}} \right)}^{2}}-\frac{4}{{{L}_{2}}{{C}_{2}}}}$; ${{P}_{4}}=-\frac{1}{2~{{R}_{2}}{{C}_{2}}}-\frac{1}{2}\sqrt{{{\left( \frac{1}{{{R}_{2}}{{C}_{2}}} \right)}^{2}}-\frac{4}{{{L}_{2}}{{C}_{2}}}}$; ${{Z}_{1}}=+\frac{j}{\sqrt{{{L}_{1}}{{C}_{1}}}}$; ${{Z}_{2}}=-\frac{j}{\sqrt{{{L}_{1}}{{C}_{1}}}}$; ${{Z}_{3}}=+\frac{j}{\sqrt{{{L}_{2}}{{C}_{2}}}}$, and ${{Z}_{4}}=-\frac{j}{\sqrt{{{L}_{2}}{{C}_{2}}}}$.         (14)

The pole and zero locations in the s-domain are graphically represented in Figure 8. Substituting s= j$\omega $ into (13), the phase of H($j\omega $) is formulated as:

$\angle H\left( j\omega  \right)=-{{\tan }^{-1}}\left[ \frac{\omega \left( \frac{{{\omega }_{1}}\omega _{2}^{2}}{{{Q}_{1}}}+\frac{{{\omega }_{2}}\omega _{1}^{2}}{{{Q}_{2}}} \right)-{{\omega }^{3}}\left( \frac{{{\omega }_{1}}}{{{Q}_{1}}}+\frac{{{\omega }_{2}}}{{{Q}_{2}}} \right)}{{{\omega }^{4}}-\left( \omega _{1}^{2}+\omega _{2}^{2} \right){{\omega }^{2}}+\omega _{1}^{2}\omega _{2}^{2}} \right]$          (15)

From (15), the phase is computed as: $\angle H\left( j\omega  \right)=$-π/2 for ω =ω₁ and $\angle H\left( j\omega  \right)=$π/2 for ω=ω₂. This analysis indicates that the cascaded filter depicted in Figure 7 can effectively notch two distinct frequencies.

This approach can be extended to notch n-number of pole frequencies. For this purpose, n HQNFs are cascaded, as illustrated in Figure 9, forming what is commonly known as a "comb filter" due to its frequency response pattern. The overall transfer function of the filter in Figure 9 is derived by multiplying the transfer functions of each cascaded HQNF, as given in Eq. (16).

In the design, the pole frequencies of the HQNFs are strategically chosen to enable the comb filter to block the power-line frequency and its odd harmonics. For instance, if the comb filter is intended to filter out the undesired power-line interference (PLI) of frequency f₁ and its odd harmonics f₂, f₃, ..., f_n, the HQNFs are designed with pole frequencies corresponding to f₁, f₂, f₃, and f_n. This configuration results in an undistorted output signal V_OUT after the comb filter rejects the undesirable frequencies.

Fabricating inductors for integrated circuits poses significant challenges due to the layer-oriented nature of most IC fabrication procedures. On-chip inductors also consume considerable chip space, and integrating passive resistance in ICs is impractical due to space constraints and power consumption. Consequently, the proposed comb filter is designed with a novel active element, the Voltage Differencing Gain Amplifier (VDGA), to streamline the implementation of the integrated circuit.

As discussed earlier, the HQNF serves as the fundamental building block of the proposed comb filter. Initially, an HQNF is devised using VDGA, as depicted in Figure 10, with its transfer function and parameters resembling those of the parallel-LC HQNF shown in Figure 5 (b).

$H\left( s \right)=\frac{{{V}_{out~}}\left( s \right)}{{{V}_{in}}\left( s \right)}=\left( \frac{{{s}^{2}}+{}^{1}/{}_{{{L}_{1}}{{C}_{1}}}}{{{s}^{2}}+{}^{s}/{}_{{{R}_{1}}{{C}_{1}}}+{}^{1}/{}_{{{L}_{1}}{{C}_{1}}}} \right)\left( \frac{{{s}^{2}}+{}^{1}/{}_{{{L}_{2}}{{C}_{2}}}}{{{s}^{2}}+{}^{s}/{}_{{{R}_{2}}{{C}_{2}}}+{}^{1}/{}_{{{L}_{2}}{{C}_{2}}}} \right)\left( \frac{{{s}^{2}}+{}^{1}/{}_{{{L}_{3}}{{C}_{3}}}}{{{s}^{2}}+{}^{s}/{}_{{{R}_{3}}{{C}_{3}}}+{}^{1}/{}_{{{L}_{3}}{{C}_{3}}}} \right)\ldots \ldots \left( \frac{{{s}^{2}}+{}^{1}/{}_{{{L}_{n}}{{C}_{n}}}}{{{s}^{2}}+{}^{s}/{}_{{{R}_{n}}{{C}_{n}}}+{}^{1}/{}_{{{L}_{n}}{{C}_{n}}}} \right)$       (16)

9.png

Figure 9. Realization of comb filter using n-number of HQNFs

10.png

Figure 10. Realization of HQNF using VDGA

To determine the transfer function, the circuit illustrated in Figure 10 is analysed as follows:

Utilising the terminal-relationship of the VDGA given in (1), the current I_Z can be expressed as ${{I}_{Z}}={{g}_{mA}}\left( {{V}_{P}}-{{V}_{N}} \right)$. Consequently, the voltage at the terminal Z (V_Z) is given by:

${{V}_{Z}}=\frac{{{I}_{Z}}}{s{{C}_{1}}}=\frac{{{g}_{mA}}\left( {{V}_{IN}}-{{V}_{OUT}} \right)}{s{{C}_{1}}}$        (17)

Again, employing (1), the current through terminal W of the VDGA can be expressed as ${{I}_{W}}={{g}_{mB}}{{V}_{Z}}-{{g}_{mC}}{{V}_{W}}$. Additionally, the current through terminal N is zero, i.e., ${{I}_{N}}$= 0. From Figure 10, the current through capacitor C2 is determined as ${{I}_{C2}}=s{{C}_{2}}\left( {{V}_{OUT}}-{{V}_{IN}} \right).$ Applying Kirchhoff’s current law (KCL) at node V_OUT yields:

${{I}_{W}}={{I}_{C2}}+{{I}_{N}}={{I}_{C2}}+0={{I}_{C2}}$        (18)

Now, (18) can be rearranged as:

${{g}_{mB}}{{V}_{Z}}-{{g}_{mC}}{{V}_{W}}=s{{C}_{2}}\left( {{V}_{OUT}}-{{V}_{IN}} \right)$         (19)

By substituting ${{V}_{Z}}$ from (17) and using ${{V}_{W}}={{V}_{OUT}}$ from Figure 10, (19) is further rewritten as:

${{g}_{mB}}\left[ \frac{{{g}_{mA}}\left( {{V}_{IN}}-{{V}_{OUT}} \right)}{s{{C}_{1}}} \right]-{{g}_{mC}}{{V}_{OUT}}=s{{C}_{2}}\left( {{V}_{OUT}}-{{V}_{IN}} \right)$           (20)

Rearranging (20) yields the transfer function of the HQNF as:

$H\left( s \right)=\frac{{{V}_{OUT}}\left( s \right)}{{{V}_{IN}}\left( s \right)}=\frac{{{s}^{2}}+\frac{{{g}_{mA}}{{g}_{mB}}}{{{C}_{1}}{{C}_{2}}}~}{{{s}^{2}}+s\frac{{{g}_{mC}}}{{{C}_{2}}}+\frac{{{g}_{mA}}{{g}_{mB}}}{{{C}_{1}}{{C}_{2}}}}$          (21)

The coefficients of the transfer functions of the VDGA-based HQNF and the passive HQNF given in (6) and (21), respectively, are compared to derive equivalent expressions for the passive elements R, L, and C of Figure 5 (b) in terms of the transconductances ((g_mA, g_mB, and g_mC) of the VDGA and the capacitors used (C₁ and C₂) in Figure 10 as follows:

$R=\frac{1}{{{g}_{mC}}}$, $L=\frac{{{C}_{1}}}{{{g}_{mA}}{{g}_{mB}}}$, and $C={{C}_{2}}$           (22)

From (22), it can be observed that the resistance R and the inductance L values depend on the value of transconductances. As these transconductances can be varied by varying the bias current of the VDGA, as given in (2) and (3), the values of R and L can be changed by adjusting the bias current of the VDGA. The pole frequency and the quality factor of the VDGA-based HQNF can be rewritten using (6) and (22) as follows:

${{f}_{0}}=\frac{1}{2\pi \sqrt{LC}}=\frac{1}{2\pi }\sqrt{\frac{{{g}_{mA}}{{g}_{mB}}}{{{C}_{1}}{{C}_{2}}}}$ and $Q=R\sqrt{\frac{C}{L}}=\frac{1}{{{g}_{mC}}}\sqrt{\frac{{{C}_{1}}{{C}_{2}}}{{{g}_{mA}}{{g}_{mB}}}}$       (23)

$H\left( s \right)=\frac{{{V}_{out~}}\left( s \right)}{{{V}_{in}}\left( s \right)}=\left( \frac{{{s}^{2}}+\frac{{{g}_{mA1}}{{g}_{mB1}}}{{{C}_{1}}{{C}_{2}}}~}{{{s}^{2}}+s\frac{{{g}_{mC1}}}{{{C}_{2}}}+\frac{{{g}_{mA1}}{{g}_{mB1}}}{{{C}_{1}}{{C}_{2}}}} \right)\left( \frac{{{s}^{2}}+\frac{{{g}_{mA2}}{{g}_{mB2}}}{{{C}_{3}}{{C}_{4}}}}{{{s}^{2}}+s\frac{{{g}_{mC2}}}{{{C}_{4}}}+\frac{{{g}_{mA2}}{{g}_{mB2}}}{{{C}_{3}}{{C}_{4}}}~} \right)\ldots \ldots \left( \frac{{{s}^{2}}+\frac{{{g}_{mAn}}{{g}_{mBn}}}{{{C}_{\left( 2n-1 \right)}}{{C}_{\left( 2n \right)}}}}{{{s}^{2}}+s\frac{{{g}_{mC2}}}{{{C}_{\left( 2n \right)}}}+\frac{{{g}_{mA2}}{{g}_{mB2}}}{{{C}_{\left( 2n-1 \right)}}{{C}_{\left( 2n \right)}}}} \right)$         (24)

11.png

Figure 11. Realization of a generalized comb filter using VDGA-based HQNF

From (23), it is evident that ${{f}_{0}}$ and $Q$ can be electronically tuned by adjusting the bias current of VDGA. Additionally, it is noteworthy that ${{f}_{0}}$ and $Q$ can be independently adjusted.

A generalized comb filter, utilizing an n-number of VDGA-based High-Quality Notch Filters (HQNF), as depicted in Figure 10, is implemented in Figure 11. This configuration is capable of blocking n number of undesirable frequencies. The transfer function of the filter circuit can be determined by multiplying the transfer functions of each cascaded VDGA-based HQNF, as given in Eq. (24).

Now, based on (24), the pole frequency (f_0n) and the quality factor (Q_n) of each HQNF can be expressed as:

${{f}_{0n}}=\frac{1}{2\pi \sqrt{{{L}_{n}}{{C}_{n}}}}=\frac{1}{2\pi }\sqrt{\frac{{{g}_{mAn}}{{g}_{mBn}}}{{{C}_{n}}{{C}_{n-1}}}}$

${{Q}_{n}}=R\sqrt{\frac{{{C}_{n}}}{{{L}_{n}}}}=\frac{1}{{{g}_{mCn}}}\sqrt{\frac{{{C}_{n}}{{C}_{n-1}}}{{{g}_{mAn}}{{g}_{mBn}}}}$        (25)

It is evident from Eq. (25) that the pole frequency, quality factor, and, consequently, the bandwidth of the nth HQNF can be adjusted by varying the transconductances (${{g}_{mAn}}$, ${{g}_{mBn}}$, and ${{g}_{mCn}}$) of their respective VDGA. This adjustment is achieved by varying the bias currents of the respective VDGA while keeping the values of capacitors ${{C}_{n}}~\text{and}~{{C}_{n-1}}$ fixed.

The performances of an ideal circuit are often deviated due to the presence of terminal parasitics in the VDGA, along with voltage and current transfer errors. These parasitics and errors represent practical imperfections that can impact the behavior of the circuit, leading to deviations from the expected ideal performance. Understanding and mitigating these non-idealities are crucial in designing and analyzing circuits to ensure they operate as intended in real-world conditions.

Considering the non-ideal transfer gains and terminal parasitics of the VDGA, the terminal relationship given in Eq. (1) is modified to account for these deviations. The modified terminal relationship is expressed as:

$\left[ \begin{matrix}

   \begin{matrix}

   {{I}_{P}}  \\

   {{I}_{N}}  \\

\end{matrix}  \\

   {{I}_{Z}}  \\

   \begin{matrix}

   \begin{matrix}

   {{I}_{X}}  \\

   {{I}_{W}}  \\

\end{matrix}  \\

   {{V}_{W}}  \\

\end{matrix}  \\

\end{matrix} \right]=\left[ \begin{matrix}

   \begin{matrix}

   0 & 0  \\

   0 & 0  \\

   {{\gamma }_{A}}{{g}_{mA}} & -{{\gamma }_{A}}{{g}_{mA}}  \\

\end{matrix} & \begin{matrix}

   0 & \begin{matrix}

   0 & 0  \\

\end{matrix}  \\

   0 & \begin{matrix}

   0 & 0  \\

\end{matrix}  \\

   0 & \begin{matrix}

   0 & 0  \\

\end{matrix}  \\

\end{matrix}  \\

   \begin{matrix}

   0 & 0  \\

\end{matrix} & {{\gamma }_{B}}\begin{matrix}

   {{g}_{mB}} & \begin{matrix}

   0 & 0  \\

\end{matrix}  \\

\end{matrix}  \\

   \begin{matrix}

   \begin{matrix}

   0  \\

   0  \\

\end{matrix} & \begin{matrix}

   0  \\

   0  \\

\end{matrix}  \\

\end{matrix} & \begin{matrix}

   \begin{matrix}

   -{{\gamma }_{B}}{{g}_{mB}}  \\

   \frac{{{\gamma }_{B}}}{{{\gamma }_{C}}}\beta   \\

\end{matrix} & \begin{matrix}

   \begin{matrix}

   0  \\

   0  \\

\end{matrix} & \begin{matrix}

   {{\gamma }_{C}}{{g}_{mC}}  \\

   0  \\

\end{matrix}  \\

\end{matrix}  \\

\end{matrix}  \\

\end{matrix} \right]\left[ \begin{matrix}

   \begin{matrix}

   {{V}_{P}}  \\

   {{V}_{N}}  \\

\end{matrix}  \\

   {{V}_{Z}}  \\

   \begin{matrix}

   {{V}_{X}}  \\

   {{V}_{W}}  \\

\end{matrix}  \\

\end{matrix} \right]$                  (26)

Here, $\beta =\frac{{{g}_{mB}}}{{{g}_{mC}}}$ and ${{\gamma }_{A}},~{{\gamma }_{B}}~{\text{and}}~{{\gamma }_{C}}$ are transconductance gain errors. Taking these errors into account, the pole frequency (f₀) and the quality factor (Q) of the proposed comb filter can be expressed as:

${{f}_{0}}=\frac{1}{2\pi \sqrt{{{L}_{n}}{{C}_{n}}}}=\frac{1}{2\pi }\sqrt{\frac{{{\gamma }_{An}}{{\gamma }_{Bn}}{{g}_{mAn}}{{g}_{mBn}}}{{{C}_{n}}{{C}_{n-1}}}}$          (27)

$Q=R\sqrt{\frac{{{C}_{n}}}{{{L}_{n}}}}=\frac{1}{{{\gamma }_{Cn}}{{g}_{mC}}}\sqrt{\frac{{{C}_{n}}{{C}_{n-1}}}{{{\gamma }_{An}}{{\gamma }_{Bn}}{{g}_{mA}}{{g}_{mB}}}}$         (28)

The sensitivities of f₀ and Q with respect to transconductance gain errors and capacitors are derived as follows:

$S_{{{\gamma }_{An}}}^{{{f}_{0}}}=S_{{{\gamma }_{Bn}}}^{{{f}_{0}}}=S_{{{g}_{mAn}}}^{{{f}_{0}}}=S_{{{g}_{mBn}}}^{{{f}_{0}}}={}^{1}/{}_{2}$        (29)

$S_{{{C}_{n}}}^{{{f}_{0}}}=S_{{{C}_{n-1}}}^{{{f}_{0}}}=-{}^{1}/{}_{2}$           (30)

$S_{{{\gamma }_{An}}}^{Q}=S_{{{\gamma }_{Bn}}}^{Q}=S_{{{g}_{mAn}}}^{Q}=S_{{{g}_{mBn}}}^{Q}={}^{1}/{}_{2}$           (31)

$S_{{{C}_{n}}}^{Q}=S_{{{C}_{n-1}}}^{Q}=-{}^{1}/{}_{2}$          (32)

$S_{{{\gamma }_{Cn}}}^{Q}=S_{{{g}_{mCn}}}^{Q}=-1$             (33)

18.png

Figure 18. Symbolic representation of a practical VDGA

19.png

Figure 19. Proposed HQNF, including parasitics

From the derived sensitivities, it is observed that the magnitude of sensitivities is within the acceptable limit of unity. To assess the performance of the real comb filter, the circuit is analyzed considering the terminal parasitics of the VDGA. The practical VDGA, including terminal parasitics, is symbolically represented in Figure 18. Utilizing this representation, the proposed High-Quality Notch Filter (HQNF) can be redrawn, as illustrated in Figure 19.

The transfer function of the practical HQNF, illustrated in Figure 19, is derived as follows:

Applying KCL at node V_OUT, the equation ${{I}_{W}}={{I}_{C2}}+{{I}_{N}}$ is obtained and can be expressed using (26) as:

${{g}_{mB}}{{V}_{Z}}-{{g}_{mC}}{{V}_{W}}={{I}_{C2}}+{{I}_{N}}$              (34)

Further, the voltage of the W-terminal, V_W, is written using Kirchhoff’s voltage law (KVL) as:

${{V}_{W}}={{V}_{OUT}}+{{I}_{W}}{{R}_{W}}$.            (35)

Combining (34) and (35), the following expression is obtained:

${{g}_{mB}}{{V}_{Z}}-{{g}_{mC}}{{V}_{OUT}}={{I}_{C2}}\left( 1+{{R}_{W}}{{g}_{mC}} \right)+{{I}_{N}}\left( 1+{{R}_{W}}{{g}_{mC}} \right)$         (36)

Rearranging (36) with the help of Figure 16, it becomes:

${{g}_{mB}}{{V}_{Z}}-{{g}_{mC}}{{V}_{OUT}}=s{{C}_{2}}\left( {{V}_{OUT}}-{{V}_{IN}} \right)\left( 1+{{R}_{W}}{{g}_{mC}} \right)+{{V}_{OUT}}\frac{\left( s{{C}_{P}}{{R}_{P}}+1 \right)}{{{R}_{P}}}\left( 1+{{R}_{W}}{{g}_{mC}} \right)$.             (37)

Again, applying KCL at terminal Z of Figure 19, the equation ${{I}_{Z}}={{I}_{C1}}+{{I}_{Z'}}$ is obtained using (26), and after rearranging, the expression for the voltage at the Z-terminal is derived as:

${{V}_{Z}}=\frac{{{g}_{mA}}\left( {{V}_{IN}}-{{V}_{OUT}} \right)}{s{{C}_{1}}+\frac{\left( s{{C}_{Z}}{{R}_{Z}}+1 \right)}{{{R}_{Z}}}}$            (38)

In summary, the transfer function derived in (40) reveals that the proposed filter may experience some impact due to the parasitics of the real VDGA. However, under conditions of low terminal parasitic capacitances, low R_W values, and high R_P and R_Z values, Eq. (40) simplifies, yielding the same transfer function as in Eq. (21). This analysis concludes that the proposed filter's non-ideal effects can be effectively suppressed within satisfactory limits by carefully selecting an appropriate architecture for the VDGA.