Low-Voltage Low-Power Filters with Independent ω₀ and Q Tuning for Electronic Cochlea Applications

Abstract

An acoustic second-order low-pass filter is proposed for filter banks emulating the operation of a human cochlea. By using a special filter structure and an innovative quality (Q)-factor tuning technique, an independent change of the cutoff frequency (ω₀) and the Q-factor with unchanged gain at low frequencies is achieved in this filter. The techniques applied result in a simple filter design with low Q-factor sensitivity to component mismatch. These filter features greatly simplify the implementation of the electronic cochlea in CMOS technologies. An exemplary filter bank designed and simulated in an X-FAB 180 nm CMOS process is presented, which consumes 1.25–34.75 nW of power per individual filter when supplied with 0.5 V. The 11-channel filter bank covers a 20–20 kHz band, while the Q-factor of each channel can be tuned from 2 to 40. The simulation-predicted sensitivities of Q and ω₀ to process/voltage/temperature (PVT) variations are less than 1%. The input-referred noise is no greater than 22 µV_RMS, and the dynamic range is at least 68 dB for all filters in the bank.

1. Introduction

Systems for effective human speech processing and recognition are of increasing practical importance. They are used in controlling machines with the human voice [1], as well as in hearing aids [2,3,4,5,6]. Electronic circuits that emulate the operation of a human cochlea are gaining interest in such applications. The popularity of this approach comes from the belief that millions of years of evolution have led to a solution that is optimally suited for processing human speech in typical human environments. Work conducted on the physiology of hearing, particularly on the functioning of the cochlea, has enabled understanding the operation of this very important organ of the inner ear [7,8,9]. Due to the special properties of the cochlea, humans are able to hear and understand quiet speech even in environments with disruptive sounds. This highly desirable feature of human hearing is related to the ability of the cochlea to analyze and dynamically adjust to the properties of the sound heard. The cochlea processes the pressure of the received acoustic wave in multiple paths. Thus, the operation of the cochlea is similar to the operation of a system with multiple bandpass channels whose sensitivity is dynamically tuned to the volume of the sound, whereas out-of-band frequency components are largely suppressed. An accurate, complex model of the cochlea [10,11] takes into account both the processing of acoustic signals in individual channels and the coupling between the channels. However, due to practical limitations in electronic VLSI (very-large-scale integration) cochlea implementations, filter banks emulating only bandpass channels without coupling are most often used as compromise solutions [12,13,14,15,16,17,18]. Two architectures of the filter banks are used, consisting of elementary sections connected in cascade [12,13,14,15] or in parallel [16,17,18]. In both architectures, the key and difficult-to-implement feature is the low-pass biquadratic filter section with electronically programmable cutoff frequency and quality factor. For precise signal processing, the frequency characteristics of biquadratic sections should be insensitive to process/voltage/temperature (PVT) variations.

A number of publications have been devoted to the design of filters intended for electronic cochlea. The earliest developed solutions [10,11,12,13,14] feature relatively high power consumption and supply voltage; therefore, they cannot be used in implants or systems with limited power supply (powered by small batteries). In addition, some of these solutions [12,13] do not provide sufficient accuracy in signal processing, which is limited by the influence of circuit parameter variations.

This paper proposes a new filter solution well suited to the requirements of an electronic cochlea powered by a small battery or small renewable energy source based on, e.g., thermoelectric generators or electromagnetic energy harvesters. By using a “transistorized” filter structure with transistors operating in the sub-threshold range [19], a significant reduction in voltage and power supply was achieved. As a result of the special tuning method of the filter quality factor and other applied techniques, a high robustness to PVT variations was obtained.

2. Proposed Filter Solution

2.1. Principle of Operation

The transfer function of the second-order low-pass filter (biquadratic) expressed in the Laplace domain is as follows:

$$H\left(s\right)=\frac{V_{out}\left(s\right)}{V_{in}\left(s\right)}=A\cdot \frac{{\omega }_0^2}{s^2+s{\omega }_0/Q_{LP}+{\omega }_0^2}$$

(1)

where V_out and V_in are the output and input voltages of the filter, A in V/V is the gain of the filter at zero frequency (DC gain), ω₀ in rad/s is the natural frequency (ω₀ = 2πf₀), and Q_LP (dimensionless) denotes a quality factor.

Figure 1 shows a family of plots of the amplitude characteristics, |H(jω)| versus ω, for three exemplary values of Q_LP. The plots are normalized for ω₀ of 1 rad/s, and A is assumed to be 1 V/V.

The bulged portion of the characteristic around ω₀ can be used for bandpass filtering if Q_LP is at least 2. In this case (Q_LP ≥ 2), ω₀ and Q_LP can be treated as a center (or peak-gain) frequency and as a quality factor of a bandpass filter, i.e., ω_center (or ω_peak) and Q_BP. The detailed numerical data show that ω₀ is practically equal to ω_peak (ω₀ ≅ ω_peak) if Q_LP is at least 2. The factor Q_BP is approximately equal to Q_LP (Q_BP ≈ Q_LP) when Q_LP is 2–4, and Q_BP is practically equal to Q_LP (Q_BP ≅ Q_LP) if Q_LP is 5 or more.

This particular shape of the characteristics allows this filter to emulate the basic properties of the cochlea in the human ear. By using an appropriate set of such filters, the functioning of the human inner ear can be imitated faithfully enough [12,13,14,15,16,17,18]. Electronic emulation of the cochlea uses filter banks in which each filter emulates particular resonant property of this organ. In simple terms, adaptation of the electronic cochlea to different sound levels involves splitting the sound spectrum into a number of channels with different center (peak-gain) frequencies, and then adjusting the gain in each channel individually. In channels where the sound strength is low, the filter gain is increased, and, in channels where the sound strength is high, the gain is reduced accordingly to achieve the most effective sound processing. To faithfully represent the physiological properties of the cochlea, the gain in the passband of each filter must be adjusted to mimic the ear’s ability to adapt to different sound levels. This capability can be achieved by tuning the quality factor Q_LP of the low-pass filter since the value of this factor determines the value of the filter peak gain, as shown in Figure 1 (|H(jω_peak)| ≈ Q_LP for Q_LP < 2, |H(jω_peak)| ≅ Q_LP for Q_LP ≥ 2).

Figure 2 shows a proposed filter that has the desirable feature of tuning the Q_LP-factor by changing the attenuation (k_Q). The filter consists of two transconductance amplifiers (G_m1, G_m2), two capacitors (C₁, C₂), and one attenuator (k_Q).

The filter in Figure 2 has the transmittance in Equation (1), in which A, ω₀ and Q_LP are

$$A=1,\hspace{1em}{\omega }_0=\sqrt{\frac{G_{m1}{G}_{m2}}{C_1{C}_2}},\hspace{1em}{Q}_{LP}=\frac{1}{k_Q}\cdot \sqrt{\frac{G_{m1}{C}_2}{G_{m2}{C}_1}}$$

(2)

where k_Q is the voltage gain (in V/V) of the attenuator. Thus, k_Q is assumed to be smaller than 1. All the undefined parameters have their usual meaning. The proposed high-Q filter structure with tunable Q is an extended version of the low-Q and not tunable structure presented in [19].

2.2. Sensitivity

In electronic cochlea emulators, banks consisting of many filters with center (peak-gain) frequencies of little difference are often used. In such solutions, it is particularly important to ensure that the peak gain frequencies and the peak gain magnitudes of the individual filters are sufficiently precise. The accuracy of filters implemented in CMOS technologies is influenced by the PVT variations. For the proposed filter in Figure 2, the influence of these factors on the peak gain frequency (≅ω₀) and the peak gain magnitude (≅Q_LP) can be determined according to the following sensitivities:

$$\left|S_{k_{G12}}^{{\omega }_0}\right|=\left|\frac{d{\omega }_0/{\omega }_0}{d{k}_{G12}/k_{G12}}\right|=0.5,\hspace{1em}\left|S_{k_{C21}}^{{\omega }_0}\right|=0.5,\hspace{1em}{S}_{k_Q}^{{\omega }_0}=0,$$

(3)

$$S_{k_{G12}}^{Q_{LP}}=\frac{d{Q}_{LP}/Q_{LP}}{d{k}_{G12}/k_{G12}}=0.5,\hspace{1em}{S}_{k_{C21}}^{Q_{LP}}=0.5,\hspace{1em}{S}_{k_Q}^{Q_{LP}}=-1,$$

(4)

where G_m1/G_m2 = k_G12, and C₂/C₁ = k_C21.

While determining the sensitivities, the peak gain frequency ω₀ is assumed to be tuned by a simultaneous change of the transconductances G_m1, G_m2 or capacitances C₁, C₂ such that their ratio G_m1/G_m2 (or C₂/C₁) is constant and, thus, Q_LP remains unchanged. However, because of technology nonidealities, these ratios may vary, especially if an ω₀ tuning range is wide. Notice that the sensitivities in Equations (3) and (4) are relatively small (≤1) and do not depend on the value of Q_LP. This means that the inaccuracy of ω₀ and Q_LP does not exceed the inaccuracy of the parameters of the technology itself, which is a very advantageous feature of this filter structure. For example, a 1% variation in the transconductance ratio k_G12 due to a technology imperfection induces only a 0.5% change in peak gain magnitude. Moreover, a high value of the peak gain magnitude can be obtained at low values of the transconductance and capacitance ratios, e.g., with k_G12 and k_C21 in the range of 1–2. Due to this, transconductors and capacitors of similar parameter values (unit components) can be used in the filter, which allows for an even better matching of these components, thereby obtaining a much better accuracy of ω₀ and Q_LP realization. In addition, the use of the unit components significantly simplifies the filter design.

2.3. Noise

The filter noise performance can be analyzed according to the schematic in Figure 3, in which the equivalent (input-referred) noise sources for each amplifier are introduced.

Assuming that the noise sources are characterized by the power spectral densities, $v_{n1}^2\left(\omega \right)$, $v_{n2}^2\left(\omega \right)$, and $v_{n3}^2\left(\omega \right)$, the respective powers of noise at the filter output are

$$\begin{array}{l}{v}_{n1,out}^2\left(\omega \right)=v_{n1}^2\left(\omega \right)\cdot {\left|H\left(j\omega \right)\right|}^2,\hspace{1em}{v}_{n2,out}^2\left(\omega \right)=v_{n2}^2\left(\omega \right)\cdot {\left|\frac{j\omega C_1}{G_{m1}}\right|}^2\cdot {\left|H\left(j\omega \right)\right|}^2,\hspace{1em}\\ v_{n3,out}^2\left(\omega \right)=v_{n3}^2\left(\omega \right)\cdot {\left|\frac{-j\omega C_1{k}_Q}{G_{m1}}\right|}^2\cdot {\left|H\left(j\omega \right)\right|}^2.\end{array}$$

(5)

Thus, the total noise power spectral density, referred to the filter input, is

$$v_{n,in}^2\left(\omega \right)=\frac{v_{n1,out}^2\left(\omega \right)+v_{n2,out}^2\left(\omega \right)+v_{n3,out}^2\left(\omega \right)}{{\left|H\left(j\omega \right)\right|}^2}=v_{n1}^2\left(\omega \right)+v_{n2}^2\left(\omega \right)\cdot {\left(\frac{\omega C_1}{G_{m1}}\right)}^2+v_{n3}^2\left(\omega \right)\cdot {\left(\frac{\omega C_1}{G_{m1}}\right)}^2\cdot k_Q^2.$$

(6)

It can be deduced from Equation (5) that the noise sources of the second transconductance amplifier and the voltage attenuator, v_n2 and v_n3, are shaped identically due to the same factor of ωC₁/G_m1. However, the attenuator’s noise is reduced further due to the factor of k_Q (k_Q is smaller than 1). As a result, the attenuator does not substantially degrade the filter noise performance.

2.4. Transistor-Level Circuit

The transistor-level implementation of the OTA-C structure of Figure 2 is shown in Figure 4. The operational transconductance amplifiers (OTAs) are composed of the transistors M₁–M₄ and 2 × M₀. The additional transistors M_SH1, M_SH2 are DC voltage shifters which improve the operating point conditions for OTAs [19]. All the transistors operate in the saturated subthreshold (weak inversion) region. The transconductances of OTAs are assumed to be identical, i.e., G_m1 = G_m2 = I_ω/(n_p∙U_T) (n_p is the subthreshold slope factor for p-channel transistors, U_T is the thermal voltage, U_T = k∙T/q). The filter capacitances are also assumed to be identical, C₁ = C₂. By using the floating capacitors (C₁/2, C₂/2), a twofold reduction in total capacitance and, thus, the area occupied by the filter was achieved [19].

The attenuator consists of transistors M₅–M₆ and the current source I_Q. Its gain depends on the transconductance of the transistors M₅ and M₆, i.e., k_Q = g_m5/(g_m5 + g_m6). For I_Q equal to zero, both transistors M₅ and M₆ conduct the same current; therefore, g_m6 = g_m5, and k_Q = 1/2. Increasing I_Q makes the current flowing through M₆ larger than that through M₅; as a result, k_Q decreases, according to the following formula:

$$k_Q=\frac{1}{2}\left(1-\frac{1}{\sqrt{1+4I_q^2/I_Q^2}}\right)\approx \frac{1}{2}\left(1-\frac{I_Q}{2I_q}\right)$$

(7)

where I_q is a quiescent (DC) current flowing through M₅ and M₆ when I_Q is zero.

For the filter in Figure 4, the transmittance parameters can be tuned independently by the currents I_ω and I_Q, as explained by Equation (7) and the following formulas:

$$A=1,\hspace{1em}{\omega }_0=\sqrt{\frac{G_{m1}{G}_{m2}}{C_1{C}_2}}=\frac{I_{\omega }}{n_p{U}_T}\cdot \frac{1}{\sqrt{C_1{C}_2}},\hspace{1em}{Q}_{LP}=\frac{1}{k_Q}\cdot \sqrt{\frac{G_{m1}{C}_2}{G_{m2}{C}_1}}=\frac{1}{k_Q}$$

(8)

where G_m1 = G_m2, C₁ = C₂, and k_Q ≤ 1/2 are assumed.

For the assumed k_Q ≤ 1/2 (Q_LP ≥ 2), ω₀ is practically equal to ω_peak, while Q_LP is well correlated with the peak gain magnitude and Q_BP, as explained in Section 2.1; thus, the designed filter satisfies the following equations:

$$A=1,\hspace{1em}{\omega }_{peak}\cong {\omega }_0,\hspace{1em}\mathrm{peak} \mathrm{gain}\cong 1/k_Q,\hspace{1em}{Q}_{BP}\approx 1/k_Q.$$

(9)

3. Filter Bank

An exemplary set of 11 filters was designed to cover the full audio band from 20 Hz to 20 kHz. Thus, the peak-gain frequencies f_peak (f_peak = ω_peak/2/π) of the consecutive filters form an octave series, i.e., 20, 40, 80, …, 10,240, and 20,480 Hz. The transistor sizes (in µm/µm) in all filters are M₀ = 10/6, M₁ = M₂ = 8/2, M₃ = M₄ = 10/10, M₅ = M₆ = 2/2, M_SH1 = 30/3, M_SH2 = 10/6. The bias current flowing through transistors M_SH1–M_SH2 is 0.25 nA. The values of the current setting the filter frequencies and the capacitances used in each filter are specified in

Table 1. Simulated ¹ post-layout parameters of the individual filters with V_DD = 0.5 V.

Target fpeak (Hz)	20	40	80	160	320	640	1280	2560	5120	10,240	20,480
Iω (nA)	0.5	0.5	1	1	2	2	4	4	8	16	16
C1/2, C2/2 (pF) 2	62.36	31.13	30.54	15.22	14.81	7.35	7.09	3.49	3.34	3.16	1.53
Power (nW) 3	1.25–3.75	1.25–3.75	2.25–4.75	2.25–4.75	4.25–6.75	4.25–6.75	8.25–10.75	8.25–10.75	16.25–18.75	32.25–34.75	32.25–34.75
Obtained fpeak (Hz) 3	18.52–19.88	38.16–40.95	74.67–80.47	157.15–159.7	298.2–319.4	595.5–639.3	1192–1276	2399–2568	4801–5121	9547–10,230	19,050–20,450

¹ CMOS process: X-FAB xh018. Transistors: 1.8 V, low-VT MOSFETs. Device models: nominal. ² Single-density (1 fF/µm²) MIM capacitor values after taking layout parasitics into account. ³ For k_Q tuned from 1/2 to 1/100 (I_Q tuned within 0–5 nA).

.

Figure 5 presents the amplitude characteristics of the filter bank for different values of parameter k_Q, i.e., for different target peak gain values. The gain nonuniformity of the filter bank is relatively small, with differences between the peak gains of individual filters of ±0.5 dB for low gains and ±1 dB for high gains. The DC gain is 0 dB in all filters (A = 1). Note also that all filters feature a high stop-band attenuation of at least 50 dB.

The detailed values of the obtained peak gain magnitude, peak gain frequency f_peak, and values of the factor Q_BP are in Figure 6a–c. These figures show zoomed-in passband portions of the amplitude characteristics for the three selected filters: for the lowest (20 Hz), middle (1.28 kHz), and highest (20.48 kHz) frequencies, respectively.

In Figure 6a–c, it can be seen that, for small and medium values, the peak gain of the filters is approximately equal to Q_BP (Q_BP ≈ peak gain), which is consistent with the theory illustrated in Figure 1. However, at large values of the peak gain, the Q tuning process becomes nonlinear, i.e., the peak gain is no longer directly proportional to 1/k_Q as shown in Equation (9). For example, in Figure 6b, the peak gain is 17.5 V/V for k_Q of 1/20 V/V, and the peak gain is 38 when k_Q is 1/50. This is caused by finite output resistance of OTAs (0.3–1.5 GΩ) which limits the maximum peak gain to 50–55 V/V. The tuning of the peak gain frequency is performed according to Equation (9), i.e., ω_peak ≅ ω₀. The results in Figure 6a–c show that, when the quality factor increases, ω_peak (f_peak) moves from left to right, approaching the target value of ω₀ (f₀) (which is consistent with the theory in Figure 1). Details of the obtained f_peak values are also given in Table 1.

Simulations which take into account technology parameter variation and component mismatch show that the peak-gain-magnitude and peak-gain-frequency variations are less than 1% (1 sigma) for all filters in the bank. Exemplary Monte Carlo simulation results for the middle-band 1.28-kHz filter are shown in Figure 6d. Details of this filter’s performance under PVT variations are provided in the Appendix A.

All the filters feature a maximum amplitude of the input sinusoidal signal of at least 86 mV_peak at 1% output THD (fundamental frequency = f₀/10). The input-referred noise integrated from 1 Hz to 2f₀ is less than 22 µV_RMS for all the filters, resulting in a filter bank dynamic range of 68 dB.

4. Discussion

The filter shown in Figure 7 is often used in many practical implementations of an electronic cochlea [12,13,14,15,20]. Therefore, this section compares the key features of this filter, which has become the preferred solution in the cochlea, with the features of the proposed new filter structure shown in Figure 2.

The filter in Figure 7 has the transmittance in Equation (1) with the following parameters:

$$A=1,\hspace{1em}{\omega }_0=\sqrt{\frac{G_{m1}{G}_{m2}}{C_1{C}_2}},\hspace{1em}{Q}_{LP}={\left[\frac{C_2}{C_1}\left(\frac{G_{m1}}{G_{m2}}-\frac{G_{m3}}{G_{m2}}\right)+1\right]}^{-1}\cdot \sqrt{\frac{G_{m1}{C}_2}{G_{m2}{C}_1}}$$

(10)

The frequency ω₀ can be tuned in the same manner as in the proposed filter, shown in Figure 2, by changing G_m1, G_m2 or C₁, C₂. To achieve ω₀ tuning independent of Q_LP, it is necessary to keep constant values of the transconductance G_m1/G_m2 and capacitance C₁/C₂ ratios. Under these conditions, Q_LP tuning requires changing the transconductance of the positive-feedback-connected amplifier (G_m3). With these assumptions, the sensitivities of Q_LP to variations in transconductance and capacitance ratios are

$$S_{k_{G12}}^{Q_{LP}}=\frac{1}{2}-\sqrt{k_{G12}{k}_{C21}}\cdot Q_{LP},\hspace{1em}{S}_{k_{C21}}^{Q_{LP}}=\frac{1}{2}-\sqrt{\frac{k_{C21}}{k_{G12}}}\cdot \left(k_{G12}-k_{G32}\right)\cdot Q_{LP},\hspace{1em}{S}_{k_{G32}}^{Q_{LP}}=\sqrt{\frac{k_{C21}}{k_{G12}}}\cdot k_{G32}\cdot Q_{LP},$$

(11)

where k_G12 and k_C21 are defined in Equation (4), and k_G32 = G_m3/G_m2.

For a practical design case, i.e., k_G12 = k_C21 = 1, the quality factor is Q_LP = 1/(2 − k_G32), and Equation (10) becomes

$$S_{k_{G12}}^{Q_{LP}}=\frac{1}{2}-Q_{LP},\hspace{1em}{S}_{k_{C21}}^{Q_{LP}}=-\left(\frac{1}{2}-Q_{LP}\right),\hspace{1em}{S}_{k_{G32}}^{Q_{LP}}=2Q_{LP}-1.$$

(12)

Comparing Equations (3) and (11), defining the sensitivities of the compared filters, it can be observed that, for all practical cases when Q_LP ≥ 2, the proposed filter has a much lower sensitivity to implementation mismatch between the transconductors (G_m1, G_m2) and the capacitors (C₁, C₂). Thus, the proposed filter is better suited for practical implementation in CMOS technologies.

5. Comparison to the State of the Art and Conclusions

Requirements for the filter banks used in electronic cochlea implementation depend on their application. In systems designed to control machines with the human voice, the values of the supply voltage and power consumption are usually less important, whereas the precision of the frequency characteristics is crucial. The most difficult criteria to meet are those related to hearing aid implants and devices for Internet of things. In such cases, low supply voltage and low power consumption must additionally be ensured due to the fact that such devices are powered from small batteries or small renewable energy sources based on, e.g., thermoelectric generators or electromagnetic energy harvesters. Therefore, in the presented comparison of the filter banks reported in the literature (

Table 2. Performance comparison of filter banks designed for electronic cochlea.

Parameter	This Work (Simulated)	[15] (Measured)	[16] (Measured)	[17] (Measured)	[18] (Measured)
Technology	0.18 µm CMOS	0.35 µm CMOS	0.35 µm CMOS	1.5 µm BiCMOS	0.18 µm CMOS
Channel count	11	64×2	1	16	64 × 2
Supply voltage	0.5 V	3.3 V	3.3 V	2.8 V	0.5 V
Power consumption per single filter	1.25–34.75 nW	NA	13–20 µW 1	112 nW–5.4 µW	0.13–100 nW 2
Filter type	OTA-C	OTA-C	Floating active inductors	OTA-C	Source follower based
Signal-to-noise ratio	68 dB @1% THD	36 dB 52 dB 3	17–27 dB for HQ 24–32 dB for LQ 4 @5% THD	57 dB IDR 5	40 dB for Q ≈ 10 55 dB for Q ≈ 1 @1% HD
f0 tuning range	20 Hz–20 kHz	100 Hz–20 kHz	31 Hz–8 kHz	200 Hz–6 kHz	8 Hz–20 kHz
f0 matching (1σ)	±1%	±6%	NA	NA	±3.3%
Q tuning range	2–40	NA	2–19	1–10	1.3–39
Q matching (1σ)	±1%	±27% for Q = 1.5	NA	NA	±15%

¹ Estimated value. ² Estimated without a programmable gain amplifier. ³ With preamp gain control. ⁴ HQ—high quality factors, LQ—low quality factors. ⁵ IDR—internal dynamic range.

), special attention was paid to these critical parameters.

In the proposed filter bank, similarly to the implementation in [18], a relatively low supply voltage of 0.5 V was used, which allowed a reduction in power consumption. Note that, in [18], an additional programmable gain amplifier was required for proper operation of the filter, whose power supply is not included in the data in Table 1. Having this fact in mind, it can be concluded that the proposed solution is more beneficial in terms of power consumption. In practical implementation of the filters in CMOS technology, an important problem is to obtain a low sensitivity of the center frequency f₀ and the quality (Q) factor to PVT variations. This issue is particularly important in systems with a large number of channels [15,17,18], where the center frequencies of consecutive channels do not differ much. For this reason, a special adjustment system is used in the implementation [15]. Such a solution complicates the whole system, increasing its area and power consumption. The data in Table 2 show that the variation of f₀ and Q in the proposed filter bank is smaller compared to the other solutions. Another very critical problem in the solution proposed in [18] is a quadratic dependence of the filter gain at low frequencies on the Q-factor value. The consequence of such a dependence is a reduction in signal dynamics. This reduction results from the fact that, e.g., a fivefold increase in the filter Q-factor leads to a 25-fold increase in the filter gain and, consequently, the output signal amplitude. To reduce the impact of this phenomenon, additional attenuators at the filter input were applied in [18]. As can be seen from Equations (1) and (2), the proposed filter is free from this unfavorable feature because the low-frequency (DC) gain is always equal to one (A = 1).