1 Introduction

Group delay is a measure of the phase response linearity with respect to frequency in systems where the information is carried by the envelope of the signal. Negative group delay (NGD) is observed in filters with a positive slope phase response. Applying a band-limited signal to its input (i.e., the frequency range of the signal is within the bandwidth where a constant group delay is realized), then the filter predictability causes the envelope of the resulting output to advance in time compared to the input signal’s envelope [13,14,15,16, 28, 29, 32]. This type of filter has been utilized in several interdisciplinary applications, including biomedical signal processing, acoustic signal processing, and chaotic circuits [7, 15, 18, 19, 30].

Negative group delay filter realizations have been already presented in the literature for both base-band and RF band [1, 4, 5, 13, 14, 16, 20,21,22,23,24, 26, 28, 31,32,33]. In this work, we focus on base-band implementations, particularly those utilizing RC or active-RC circuits [1, 20, 24]. The topologies introduced in [20, 24, 25] are passive RC negative group delay filter designs, which suffer from the absence of cascadability and require additional buffer stages for interconnection. This obstacle has been overcome by the topologies in [12], where first- and second-order NGD filters realized using operational amplifiers (op-amps) as active elements have been presented. The same feature is offered by the topology in [1], where an active-RC second-order filter has been introduced with current feedback operational amplifiers (CFOAs) as active elements. The realized transfer function is an inverse band-pass filter function, which, however, has the limitation of very low gain at low and high frequencies. The second-order topologies introduced in [5] employ also CFOAs as the active elements, and they do not offer a high-impedance input node.

The contribution of this work involves a systematic study of the fundamental negative group delay filter functions by simultaneously considering their gain, phase, and group delay behavior, in order to exploit the practical restrictions. The derivations are accompanied by suitable active-RC implementations evaluated through both simulations and experimental results.

This work is organized as follows: the basic transfer functions of negative group delay filters are presented in Sect. 2, while possible implementations are provided in Sect. 3. The behavior of the considered schemes is evaluated in Sect. 4 through simulation results using the OrCAD PSpice suite and through experimental results using the AN231E04 FPAA platform from Anadigm [2].

2 Negative Group Delay Filters

2.1 First-Order Negative Group Delay Filters

An inverse low-pass filter is described by the transfer function

$$\begin{aligned} H(s)=G_{L}(\tau s+1), \end{aligned}$$
(1)

with \(\tau \) being a time constant related to the frequency of the zero (\(\omega _{z}\)) of the transfer function, i.e. \(\omega _{z}=1/\tau \) [11, 16, 28]. This is also equal to the frequency where a 3dB rise of the gain from its low-frequency value is observed and is usually known in the literature as the knee frequency \(\omega _{k}\). \(G_{L}\) is the low-frequency gain of the filter. This transfer function has been considered in [11, 16, 28]. The gain and phase responses of (1) are, respectively, given by:

$$\begin{aligned}{} & {} \mid H(j\omega )\mid =\sqrt{1+(\omega \tau )^{2}}, \end{aligned}$$
(2a)
$$\begin{aligned}{} & {} \angle H(j\omega )=\tan ^{-1}(\omega \tau ). \end{aligned}$$
(2b)

Therefore, the group delay of the filter is given by:

$$\begin{aligned} \tau _{g}(\omega )\equiv -\frac{\partial }{\partial \omega }\angle H(j\omega )=-\frac{\tau }{1+(\omega \tau )^{2}}. \end{aligned}$$
(3)

According to (3), the group delay is equal to \(-\tau \) at low frequencies, while it approaches zero at high frequencies because the bandwidth of the group delay frequency response is determined by the frequency at which \(\tau _{g}(\omega )=0\), and consequently, it has a theoretically infinite value. It is worth noting that this is an unrealistic consideration because the gain of the filter increases with the frequency and becomes infinite at high frequencies, causing the saturation of the system’s output. Therefore, a more realistic definition should be also based on the consideration of the gain of the filter. In particular, assuming that the group delay response is negative within a frequency range of interest, then the “effective” bandwidth is determined by the frequency where a 3dB rise of the gain from its minimum value is observed. For this type of filter, it can be mathematically expressed as:

$$\begin{aligned} \textrm{BW}_{\textrm{GD}}=\omega _{z}=\frac{1}{\tau }. \end{aligned}$$
(4)

A similar problem arises with the gain in the case of the inverse high-pass filter function in (5)

$$\begin{aligned} H(s)=G_{H}\frac{\tau s+1}{\tau s}, \end{aligned}$$
(5)

where a pole located at zero frequency is added compared to the transfer function in (1). This transfer function has been considered in [23], and since the phase response includes an additional term of \(-\pi /2\) compared to (2b), the realized group delay is also described by (3). As the gain of the filter is extremely high at low frequencies (where negative group delay is achieved), it makes this type of filter useless for implementing negative group delay circuits in practice.

Let us now consider the transfer function of a first-order bilinear filter

$$\begin{aligned} H(s)=G_{L}\frac{y\tau s+1}{x\tau s+1}, \end{aligned}$$
(6)

with \(x,y>0\) being dimensionless scaling factors and the time constant \(\tau =1/\omega _{0}\), where \(\omega _{0}\) is a characteristic frequency. The transfer function in (6) can be alternatively written as in (7)

$$\begin{aligned} H(s)=G_{L}\left( \frac{\omega _{p}}{\omega _{z}}\right) \frac{s+\omega _{z}}{s+\omega _{p}}, \end{aligned}$$
(7)

with the pole (\(\omega _{p}\)) and the zero (\(\omega _{z}\)) frequencies (located in the left-half of the s-plane) given by

$$\begin{aligned} \omega _{p}=\frac{1}{x\tau }=\frac{\omega _{0}}{x},\,\,\,\,\,\,\,\omega _{z}=\frac{1}{y\tau }=\frac{\omega _{0}}{y}, \end{aligned}$$
(8)

both dependent on the values of x and y variables. The magnitude and phase responses of this filter are, respectively, given by (9a)–(9b)

$$\begin{aligned}{} & {} \mid H(j\omega )\mid =G_{L}\cdot \sqrt{\frac{1+(y\omega \tau )^{2}}{1+(x\omega \tau )^{2}}}, \end{aligned}$$
(9a)
$$\begin{aligned}{} & {} \angle H(j\omega )=\tan ^{-1}(y\omega \tau )-\tan ^{-1}(x\omega \tau ), \end{aligned}$$
(9b)

and the group delay is hence

$$\begin{aligned} \tau _{g}(\omega )=-(y-x)\tau \frac{[1-(\omega \tau )^{2}xy]}{[1+(y\omega \tau )^{2}][1+(x\omega \tau )^{2}]}. \end{aligned}$$
(10)

Defining as mean frequency the geometric mean of the pole and zero frequencies, i.e.,

$$\begin{aligned} \omega _{m}\equiv \sqrt{\omega _{p}\omega _{z}}, \end{aligned}$$
(11)

the numerator of (10) becomes zero at \(\omega =\omega _{m}\), and for other frequencies the sign of the numerator changes from positive to negative. For \(x<y\) and \(\omega <\omega _{m}\), the group delay is negative and at very low frequencies tends to \(-(y-x)\tau \), having a low-pass filter behavior.

Following the previous definition, the bandwidth of the filter is determined taking into account that the gain at high frequencies is expressed by the formula \(G_{H}=G_{L}\left( y/x\right) \) and that at the mean frequency it is equal to the geometric mean of \(G_{L}\) and \(G_{H}\), i.e.,

$$\begin{aligned} G_{m}=\sqrt{G_{L}G_{H}}=G_{L}\sqrt{\frac{y}{x}}. \end{aligned}$$
(12)

In order for this frequency to be equal to the knee frequency, the condition \(G_{m}=\sqrt{2}G_{L}\) must be fulfilled which consequently means that \(y=2x\), i.e., the restriction \(x<y\) is always satisfied. For comparison purposes with the case of the first-order inverse low-pass filter, the following values are chosen: \(y=1\) and \(x=0.5\). Using (11), the resulting expression of the bandwidth is given by: (13) as

$$\begin{aligned} \textrm{BW}_{\textrm{GD}}=\omega _{m}=\frac{\sqrt{2}}{\tau }. \end{aligned}$$
(13)

According to (4) and (13), the realized bandwidth is \(\sqrt{2}\) times the corresponding bandwidth of the first-order inverse low-pass filter.

2.2 Second-Order Negative Group Delay Filters

Let us consider the transfer function of an inverse second-order low-pass filter

$$\begin{aligned} H(s)=G_{L}\frac{s^{2}+\frac{\omega _{0}}{Q}s+\omega _{0}^{2}}{\omega _{0}^{2}}. \end{aligned}$$
(14)

The zeros of (14) are

$$\begin{aligned} \omega _{z1,2}=-\frac{\omega _{0}}{2Q}\left( 1\pm \sqrt{1-4Q^{2}}\right) , \end{aligned}$$
(15)

and the gain and phase responses are, respectively,

$$\begin{aligned}{} & {} \mid H(j\omega )\mid =G_{L}\cdot \sqrt{\left[ 1-(\omega \tau )^{2}\right] ^{2}+\left( \frac{\omega \tau }{Q}\right) ^{2}}, \end{aligned}$$
(16a)
$$\begin{aligned}{} & {} \angle H(j\omega )=\tan ^{-1}\frac{\omega \tau }{Q\left[ 1-\left( \omega \tau \right) ^{2}\right] }. \end{aligned}$$
(16b)

Therefore, the expression of the group delay is:

$$\begin{aligned} \tau _{g}(\omega )=-\frac{\tau }{Q}\frac{1+(\omega \tau )^{2}}{\left[ 1-(\omega \tau )^{2}\right] ^{2}+\left( \frac{\omega \tau }{Q}\right) ^{2}}. \end{aligned}$$
(17)

According to (17), the filter achieves a negative group delay that approaches \(-\tau /Q\) at low frequencies and is dependent on the quality factor of the poles. Assuming a gain response without peaking (i.e., \(Q\le 1/\sqrt{2}\)), the (absolute) value achieved is greater than that offered by its first-order counterpart. In terms of bandwidth, in the case of a Butterworth filter function (\(Q=1/\sqrt{2}\)), the expression in (4) is still valid.

In the case of second-order inverse high-pass filter described by

$$\begin{aligned} H(s)=G_{H}\cdot \frac{s^{2}+\frac{\omega _{0}}{Q}s+\omega _{0}^{2}}{s^{2}}, \end{aligned}$$
(18)

its behavior in terms of group delay is still described by (17), but faces the same obstacle as its first-order counterpart. The transfer function of a second-order inverse band-pass filter is:

$$\begin{aligned} H(s)=G_{0}\cdot \frac{s^{2}+\frac{\omega _{0}}{Q}s+\omega _{0}^{2}}{\frac{\omega _{0}}{Q}s}, \end{aligned}$$
(19)

and its group delay is also described by (17). The problem with this band-pass filter is that the gain exhibits very high values at both low and high frequencies, limiting the circuit’s usefulness to a narrow frequency range. It must be mentioned at this point that such systematic comparison of the behavior of the transfer functions in (14), (18), and (19) does not exist in the literature. For further clarity, and considering that \(\omega _{0}=1\) rad/s, \(G_{L}=1\), \(x=0.5,\,y=1\), and \(Q=1/\sqrt{2}\), the group delay responses of the filters described by (1), (6), and (14) are depicted in Fig. 1.

Fig. 1
figure 1

Group delay responses of the filters described by (1), (6), and (14) (\(\omega _{0}=1\) \(rad/s\), \(G_{L}=1\), \(x=0.5,\,y=1\) and \(Q=1/\sqrt{2}\))

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2.3 Non-Integer-Order Filters

Considering the case of non-integer-order filters, the transfer functions of the fractional-order and power-law inverse low-pass filters are: \(H(s)=G_{L}\left[ \left( \tau s\right) ^{\alpha }+1\right] \) and \(H(s)=G_{L}(\tau s+1)^{\beta }\), respectively, with \(0<\alpha ,\beta <1\) being their orders [10]. The corresponding expressions of the realized group delays are:

$$\begin{aligned} \tau _{g}(\omega )= & {} -\alpha \tau \frac{\sin (0.5\pi \alpha )}{\left( \omega \tau \right) ^{1-\alpha }\left[ 1+\left( \omega \tau \right) ^{2\alpha }+2\left( \omega \tau \right) ^{\alpha }\cos (0.5\pi \alpha )\right] }, \end{aligned}$$
(20)
$$\begin{aligned} \tau _{g}(\omega )= & {} -\frac{\beta \tau }{1+(\omega \tau )^{2}}, \end{aligned}$$
(21)

both being negative. According to (20), at low frequencies the group delay is not constant and, also, takes extremely large values; this results from the fact that in this range \(t_{g}(\omega )\sim 1/\left( \omega \tau \right) ^{1-\alpha }\). Therefore, since it does not offer a constant negative group delay over the range of interest, this transfer function is useless in practice. On the other hand, the group delay realized by a power-law filter is constant at the low-frequency range and its value, resulting from (21) is equal to \(-\beta \tau \). This is \(\beta \) times the corresponding group delay realized by a first-order inverse low-pass filter. Taking into account that the knee frequency of the filter is given by the formula: \(\omega _{k}=\omega _{z}\sqrt{2^{1/\beta }-1}\), the achieved bandwidth will be:

$$\begin{aligned} \textrm{BW}_{\textrm{GD}}=\omega _{k}=\frac{1}{\tau }\sqrt{2^{1/\beta }-1}, \end{aligned}$$
(22)

which is greater than that realized by its integer-order counterpart, which is given by (4). Assuming, for example, that \(\beta =0.1\), then the bandwidth becomes about 32 times the bandwidth offered by the corresponding integer-order filter.

3 Realizations of Negative Group Delay Filter Functions

Employing a CFOA as the active element, the implementation of a first-order inverse low-pass filter is given in Fig. 2a, with the realized transfer function being

$$\begin{aligned} H(s)=\frac{R_{2}}{R_{1}}(R_{1}Cs+1). \end{aligned}$$
(23)

The implementation of (23) using operational amplifiers as active elements was initially presented in [16] with a low-frequency gain equal to one, but this topology suffers from the presence of a low-impedance input node. By contrast, in the topology of Fig. 2a, the input node offers a high-impedance due to the properties of the CFOA terminals. Comparing (1) with (23), the resulting design equations are: \(G_{L}=R_{2}/R_{1}\) and \(\tau =R_{1}C_{1}\).

Fig. 2
figure 2

Realizations of negative group delay using first-order a inverse low-pass filter, b bilinear filter and c second-order inverse low-pass filter

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Taking into account the effect of the parasitic resistance (\(r_{X}\)) at the X terminal of the CFOA, the resulting transfer function is

$$\begin{aligned} H\,'(s)=H(s)\frac{R_{1}}{R_{1}+r_{X}}\frac{1}{\tau _{X}s+1}, \end{aligned}$$
(24)

with H(s) being the transfer function which corresponds to the ideal case, and the extra time constant established by the parasitic resistance being

$$\begin{aligned} \tau _{X}=\frac{R_{1}r_{X}C_{1}}{R_{1}+r_{X}}=\tau \frac{r_{X}}{R_{1}+r_{X}}. \end{aligned}$$
(25)

The realized group delay function is thus

$$\begin{aligned} \tau \,'_{g}(\omega )=-\tau \frac{R_{1}}{R_{1}+r_{X}}\frac{1-(\omega \tau )^{2}\frac{r_{X}}{R_{1}+r_{X}}}{\left[ 1+(\omega \tau )^{2}\right] \cdot \left[ 1+(\omega \tau _{X})^{2}\right] }. \end{aligned}$$
(26)

According to (26), the group delay is negative under the condition

$$\begin{aligned} \omega <\frac{1}{\tau }\left( 1+\frac{R_{1}}{r_{X}}\right) . \end{aligned}$$
(27)

Taking into account that the bandwidth of the group delay in this case is equal to \(1/\tau \), it is readily obtained from (27) that the presence of \(r_{X}\) does not affect the realized bandwidth. The value of the group delay at very low frequencies is equal to \(R_{1}/(R_{1}+r_{X})\); considering a typical value of \(R_{1}\) equal to 10 k\(\Omega \) and taking into account that the parasitic resistance \(r_{X}\) of the AD844 is equal to 50 \(\Omega \), it is obvious that the effect of this resistance is negligible.

Meanwhile, the bilinear filter realization is proposed in Fig. 2b, with the realized transfer function being

$$\begin{aligned} H(s)=\frac{R_{3}}{R_{2}}\cdot \frac{(R_{1}+R_{2})C_{1}s+1}{R_{1}C_{1}s+1}. \end{aligned}$$
(28)

Assuming that \(\tau =(R_{1}+R_{2})C_{1}\), then comparing (6) and (28), it is derived that \(G_{L}=R_{3}/R_{2}\), \(x=R_{1}/(R_{1}+R_{2})\), and \(y=1\). The gain at high frequencies is given by the expression: \(G_{H}=1+(R_{2}/R_{1})\).

With regards to the effect of the parasitic resistance \(r_{X}\), the associated transfer function is:

$$\begin{aligned} H\,'(s)=\frac{R_{3}}{R_{2}+r_{X}}\cdot \frac{(R_{1}+R_{2})C_{1}s+1}{\frac{R_{1}R_{2}+r_{X}(R_{1}+R_{2})}{R_{2}+r_{X}}C_{1}s+1}. \end{aligned}$$
(29)

Inspecting (28)–(29), it is concluded that the presence of \(r_{X}\) changes the pole of the transfer function from its initial value: \(\omega _{p}=1/R_{1}C_{1}\) to a value given by (30)

$$\begin{aligned} \omega \,'_{p}=\frac{R_{2}+r_{X}}{R_{2}+r_{X}\left( 1+\frac{R_{2}}{R_{1}}\right) }\cdot \omega _{p}\equiv k\cdot \omega _{p}. \end{aligned}$$
(30)

The new location of the pole affects the factor x, which now becomes equal to \(x\,'=kx\) and, consequently, affects the value of the mean frequency (\(\omega _{m}\)), and also, the value of the \(\textrm{BW}_{\textrm{GD}}\) by a factor equal to \(\sqrt{k}\). At very low frequencies, the group delay will tend to the value \(-(y-kx)\tau \). In practice, as the values of \(R_{1}\) and \(R_{2}\) are in the order of \(k\Omega \) and \(r_{X}=50\,\Omega \), the parasitic resistance does not cause significant deviation from the theoretically predicted values.

A possible implementation of a second-order inverse low-pass filter is demonstrated in Fig. 2c [8]. Its transfer function is

$$\begin{aligned} H(s)=\frac{R_{3}}{R_{2}}\cdot \frac{s^{2}+\frac{1}{R_{2}C_{2}}s+\frac{1}{R_{1}R_{2}C_{1}C_{2}}}{\frac{1}{R_{1}R_{2}C_{1}C_{2}}}. \end{aligned}$$
(31)

Employing (14), the resulting design equations are \(G_{L}\!=\!R_{3}/R_{2}\), \(\omega _{0}\!=\!(R_{1}R_{2}C_{1}C_{2})^{-1/2}\) and \(\omega _{0}/Q=(R_{2}C_{2})^{-1}\). It is worth mentioning that a passive version of the topology in Fig. 2c is employed in [20]. However, its drawbacks include a \(-6\) dB gain in the pass band and a maximum possible gain of 0 dB.

The transfer function in (6) can be easily realized using the Anadigm AN231E04 FPAA device [2, 9], where the bilinear filter is formed by the summation of a unity gain low-pass filter with pole frequency equal to \(\omega _{0}/2\) and a high-pass filter with the same pole frequency and a maximum gain equal to 2, ensuring that the zero of the realized function will be at \(\omega _{0}\). The clock frequency of the FPAA was set equal to 500 kHz.

4 Simulation and Experimental Results

4.1 Simulation Results

Assuming, for example, \(\omega _{0}=10\,\mathrm{krad/s}\) and \(G_{L}=1\), the values of the passive elements (rounded to the E96 series defined in IEC 60063 standard) in the case of the inverse filter in Fig. 2a are \(R_{1}=R_{2}=10\,\textrm{k}\Omega \), and \(C_{1}=10\,\textrm{nF}\). The values of passive elements, which are required for implementing the transfer function in (6) by the topology in Fig. 2b with \(x=0.5=y/2\), \(G_{L}=1\) and \(\omega _{0}=10\,\mathrm{krad/s}\), are calculated as \(R_{1}=R_{2}=R_{3}=10\,\textrm{k}\Omega \), and \(C_{1}=5\,\textrm{nF}\). Considering the topology in Fig. 2c and assuming that \(R_{1}=R_{2}=10\,\textrm{k}\Omega \), the other component values for \(\omega _{0}=10\,\mathrm{krad/s}\) and \(Q=1/\sqrt{2}\) are \(R_{3}=10\,\textrm{k}\Omega ,\,C_{1}=14\,\textrm{nF}\), and \(C_{2}=7.15\) nF.

Fig. 3
figure 3

Simulated responses of the group delays realized by the circuits in Fig. 2a–c

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Fig. 4
figure 4

Monte Carlo analysis results of the low-frequency group delay and bandwidth realized by the topology in Fig. 2b

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Fig. 5
figure 5

Time-domain output waveforms of the topologies in Fig. 2a–c stimulated by a Gaussian pulse

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Fig. 6
figure 6

FPAA experimental results showing a the measured magnitude and phase of the transfer function (6) and the observed input and output waveforms on the oscilloscope using b a Gaussian mono-pulse input and c a Sync input

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Utilizing OrCAD PSpice, with the AD844 CFOA biased at ±10V [3], the simulated group delay responses for the filters of Fig. 2a–c are depicted in Fig. 3, with the associated theoretical responses represented by dashed lines. At low frequencies, the group delay values for the inverse first-order filter, the bilinear first-order filter, and the inverse second-order filter are \(-99.6\upmu \)s, \(-49.8\,\upmu \)s, and \(-139.6\,\upmu \)s, close to the theoretical ones of \(-100\,\upmu \)s, \(-50\,\upmu \)s, and \(-141.42\,\upmu \)s, respectively. The associated bandwidths are 10.15 krad/s, 14.17 krad/s and 10.16 krad/s, which are also close to those predicted by the theory, i.e., 10 krad/s, 14.14 krad/s and 10 krad/s.

Sensitivity analysis is performed by employing the Monte Carlo analysis, offered by the Advanced Analysis tool of OrCAD. The statistical plots (for \(N=500\) runs) of the low-frequency group delay and bandwidth of the first-order bilinear filter of Fig. 2b are demonstrated in Fig. 4. The values of the standard deviation are \(1.39\,\upmu \)s for the group delay and 0.4 krad/s for the bandwidth, which are derived by considering \(\pm 5\%\) deviation from the nominal values of passive elements. Given that the mean values are \(-49.8\,\upmu s\) and 14.17 krad/s, this confirms the filter’s reasonable sensitivity characteristics.

The time-domain behavior of the topologies under consideration is evaluated using as a stimulus a Gaussian pulse. The resulting output waveforms are depicted in Fig. 5 where the measured delays between the peaks of the envelop of the output signals and the input signal are \(-90.2\,\upmu \)s, \(-41.8\,\upmu \)s, and \(-151.3\,\upmu \)s closely matching the theoretical values of \(-89.6\,\upmu \)s, \(-42\,\upmu \)s and \(-164.2\,\upmu \)s, respectively.

4.2 Experimental Results

The experimental gain and phase frequency responses obtained from the FPAA setup and measured using an HP4395A network/spectrum analyzer are shown in Fig. 6a. The measured bandwidth is 15.63 krad/s, while the theoretical value is 14.17 krad/s. Using the Keysight IntuiLink Waveform Editor for Arbitrary Waveform Generators®, a Gaussian mono-pulse input and a Sync input were used as stimulus signals applied through an Agilent 33250A arbitrary waveform generator. The associated screenshots, showing the input and output waveforms on an Agilent DSO 6034A oscilloscope, are depicted in Fig. 6b, c. The measured delays between the peaks of the envelop of the output and input signals were \(-30\,\upmu \)s and \(-40\,\upmu \)s, close to the corresponding theoretical values of \(-33.2\,\upmu \)s and \(-43.35\,\upmu \)s.

5 Conclusion

The study performed in this work demonstrated another important application of inverse filters, complementing those already explored in [17, 27] for realizing a negative group delay. The same holds true for bilinear filters, which have found applications in acoustics as “shelving filters” and in control systems as lead compensators. The suitability of these filters has been approached from a filter design viewpoint where both the gain and phase response were taken into account. Future research steps include the realization of non-integer-order negative group delay filters [6].