OTRA-Based Multi-Function Inverse Filter Configuration

A new OTRA-based multifunction inverse filter configuration is presented which is capable of realizing low pass, high pass and band pass filters using only two OTRAs and five to six passive elements. To the best knowledge of the authors, any inverse filter configuration using OTRAs has not been reported in the literature earlier. The effect of the major parasitics of the OTRAs and their effect on the performance of the filter have been investigated and measured through simulation results and Monte-Carlo analysis. The workability of the proposed circuits has been confirmed by SPICE simulations using CMOS-based-OTRA realizable in 0.18 μm CMOS technology. The proposed circuits are the only ones, which provide simultaneously the following features: use of reasonable number of active elements (only 2), realizability of all the three basic filter functions, employment of all virtually grounded resistors and capacitors and tunability of all filter parameters (except gain factor, H0 for inverse high pass). The centre/cut-off frequency of the various filter circuits lying in the vicinity of 1 MHz have been found to be realizable, which has been verified through SPICE simulation results and have been found to be in good agreement with the theoretical results.


Introduction
Inverse filters are important from the view point of some applications in the areas of Communication, Control and Instrumentation Systems where the distortion of the signal, caused by the signal processor or transmission systems, can be corrected by inverse filters, the frequency response of which is the reciprocal of the frequency response of the signal processors or transmission systems which were responsible for creating the undesired distortion in the system.
In [1], a general method for obtaining inverse transfer function for linear dynamic systems and the inverse transfer characteristics of nonlinear resistive systems using nullors as basic building block were presented.In [2], a procedure for transforming voltage-mode opamp-based RC filter into a current mode, Four Terminal Floating Nullors (FTFN)-based inverse filter was given.In [3], another general procedure for the realization of FTFN-based inverse filter from the welldeveloped voltage-mode filters was presented.In [4], Abuelma'atti used a single FTFN to realize many filters including an inverse filter.In [5], Gupta, Bhaskar, Senani and Singh presented four new configurations which realize inverse low pass, band pass, high pass and band reject filters using AD844-type Current Feedback Operational Amplifiers (CFOA).In [6], Gupta, Bhaskar and Senani presented a number of new configurations that realize inverse LP, HP, and BR filters using CFOAs which offer the properties superior to the previously known inverse filters.In [7], a Current-Differencing Buffered Amplifier (CDBA)-based universal inverse filter which realizes five types of inverse filters was proposed.In [8], a general configuration, derived from the circuit of Fig. 1 of [5], was presented by Wang, Chang, Yang and Tsai from which low pass, band pass and high pass inverse filters were shown to be realizable by appropriate choices (resistive or capacitive) of the various circuit admittances.In [9] and [10] inverse all-pass filters using Current-Differencing Transconductance Amplifier (CDTA) and CCII were proposed.In [11] Herenscar, Lahiri, Koton and Vrba presented a configuration using three Differential Difference Current Conveyors (DDCCs), which realizes inverse low pass, band pass and high pass filter.In [12] Garg, Bhagat and Jain presented a novel multifunction inverse filter using three modified CFOAs, which normally required nine CFOAs.In [13], Leuciuc proposed a method for realization of inverse systems for circuits employing nullor for applications in chaos synchronization.In [14] Tsukutani, Sumi and Yabuki presented realization of inverse low pass, high pass and band pass filters using five/six Multiple Output Operational Transconductance Amplifiers (MOTAs) in operating both voltage and current-mode.In [15], Patil and Sharma have reported inverse filter realized using two CFOAs.In [17], Yuce, Tokat, Minaei and Cicekoglu reported low-component count insensitive current and voltage-mode PID using three current-conveyors, PI and PD controllers employing two currentconveyors.Nasir and Ahmed in [18] have presented multi function inverse filter realization using two CD-BAs.
From the survey of the existing literature as detailed above, it has been found that so far, no universal inverse filters have been proposed using the Operational Transresistance Amplifiers (OTRAs) as an active element.Therefore, the main aim of this paper is to propose a generalized universal inverse filter configuration using OTRAs which can realize low pass, high pass and band pass filters from the same configuration as special cases.

A Brief Overview of OTRA and Its Applications
The OTRA which is a differential Current-Controlled Voltage-Source (CCVS), can be symbolically shown as in Fig. 1 and is characterized by the following terminal equation and matrix characterization: The OTRA has received prominent attention in analog circuit literature for more than two decades because of its important property of providing a virtual ground at both of its input terminals, which eliminates the effect of parasitics at both the input terminals.On the other hand, its low output impedance enables the OTRA-based circuits to be cascaded easily.As a consequence, the use of OTRA has been investigated in the realization of numerous functional circuits so far such as biquad filters [19] and [20], immittance simulators [23], [24], [25], [26], [39] and [42] oscillators [21], [22] and [23], square/triangular waveform generators [27] and [29], monostable/astable multivibrators [28], all pass filters [32] and [33], analog multiplier/divider [30], PID controller [40] and [41], etc.Several CMOS implementations of the OTRAs have also been advanced in the literature, for instance, see [34], [35], [36], [37] and [38] and the references cited therein.On the other hand, an implementation of the OTRA using two AD844-type CFOAs has also been popularly employed by several researchers.

The Proposed Configuration
The proposed new generalized structure for realizing inverse low-pass, inverse high-pass and band-pass filters as special cases, using two OTRAs and five admittances is shown in Fig. 2.
From a straightforward analysis, the transfer function of the circuit of Fig. 2 has been found to be: If we choose the various admittances as Fig. 2: The proposed generalized structure for realizing inverse filters.
resulting transfer function as given by: which represents an inverse low pass filter with various parameters given by: If the admittances are chosen, such that then we obtain the transfer function that represents an inverse band pass filter: The Inverse band pass filter parameters are given by: Lastly, if the admittances are chosen as then the resulting transfer function represents the realization of an inverse high pass filter given by: The various parameter values in this case are given by: From Eq. ( 5), Eq. ( 7) and Eq. ( 9), it can be seen that most of the relevant parameters of the frequency response of the inverse low pass, band pass and high pass filters, can be independently adjusted through various resistances.Since all the circuits resulting from the proposed structure employ virtual grounded resistors and capacitors, they are suitable from the view point of IC implementation.

Non-Ideal Analysis
Ideally, the transresistance gain R m (s) of an OTRA approaches infinity and forces the two input currents to be equal.However, practically, the transresistance gain R m (s) is finite and its effect therefore needs to be considered.The one-pole model of the OTRA is given by: For middle frequencies, the transresistance gain R m (s) can be approximated as: where, R mo is the DC open-loop transresistance gain, ω o is the transresistance cut-off frequency and C p is the parasitic capacitance (the parasitic capacitance C p appears at the high impedance node between the output of the Current Differencing Unit and the input of the voltage buffer of the OTRA architecture; for instance see node 'N ' in the CMOS OTRA of Fig. 3).On the other hand, the more elaborate two-pole model of the OTRA is normally expressed as: which can be can be modified and rewritten as: In this case also, for frequencies lying in the range: ω p1 << ω << ω p2 , Eq. ( 13) can be approximated as: A straightforward analysis of the proposed generalized structure of Fig. 2, using Eq. ( 11) results in the non-ideal transfer function of the circuit given by the following expression: If we choose the admittances as , we obtain the non-ideal transfer function as: Neglecting the terms involving s 2 and s (since in general, the product C 1 C 2 would be much larger than C 2 p and C p ), we can approximate the transfer function of Eq. ( 16) as: The non-ideal inverse low pass filter parameters are given by: From the above expressions, we conclude that the non-ideal cut-off frequency and the gain are not affected due to the non-ideal behavior of OTRA but the Q 0 is affected and the non-ideal Q 0 is given as: From the above, the percentage error in Q 0 is given by: Similarly, if the admittances are chosen, such that , then we obtain the non-ideal transfer function of the inverse band pass filter given by: From Eq. ( 21) the non-ideal inverse band pass filter parameters are given by: ω0 = 1 Thus, in this case only centre frequency is affected by parasitics capacitance of the OTRA.The above parameters can be further expressed as: Hence, the percentage errors in the radian frequency and Q factor are found to be as follows: c 2017 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING Lastly, if the admittances are chosen such that then we obtain the non-ideal transfer function of the inverse high pass filter which can be expressed as: Neglecting the term C 2 p we obtain the resulting transfer function as: The non-ideal parameters of the inverse high pass filter are, therefore, given by: ω0 = 1 Thus, the non-ideal parameters can be expressed in terms of ideal parameters, as follows: Hence, percentage errors in the filter parameters are found to be: % error in radian frequency will be: % error in the Q factor would be: Lastly, the % error in the gain will be:

Comparison With Previously Published Inverse Filters
A comparative study of all the inverse filters realized using various active elements is shown in Tab. 1. From the table, a comparison of the salient features of the proposed circuits with those previously known [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [14], [15], [16] and [18] reveals that the proposed circuit configuration is the only one which provides simultaneously the following features: use of reasonable number of active elements (only 2), realizability of all the three basic filter functions, employment of all virtually grounded resistors and capacitors and tunability of all filter parameters (except gain factor, H 0 for inverse high pass).

SPICE Simulation and Experimental Results
The workability of the all the three special cases of the proposed configuration has been verified through SPICE simulations using CMOS OTRA as presented by Mostafa-Soliman in 2006 [31] (reproduced here in Fig. 3 with the aspect ratios of MOSFETs as given in Tab. 1 and the model parameters using 0.18 µm CMOS technology provided by TSMC as given in Tab. 3. The SPICE simulation results of Fig. 5, Fig. 6 and Fig. 7 are seen to be consistent with the similar results obtained by other inverse filters using different devices known earlier [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], Fig. 4: OTRA implemented using two commercially available AD844 type CFOA ICs [22] and [27].
Tab. 3: Aspect ratios of the various MOSFETs for the circuit of Fig.
The percentage error between theoretical and the simulated results have been found to be nearly of the same order as those calculated.

SECTION POLICIES
CenterFreq(V(11),1) Voltage gain (dB) In conclusion, a good correspondence has been found between theoretical results and SPICE results as well as between theoretical results and experimental results.

Conclusion
Among numerous alternative building blocks investigated in analog circuit literature over the last three the OTRA has been prominently employed in the realization of numerous analog signal processing/signal generation applications in recent years.However, the use of OTRA in the realization of inverse filters had not been investigated earlier.
This paper fills this void by proposing a two-OTRAbased generalized configuration from which low-pass, band-pass and high-pass filters can be realized as specases by judicious choices (resistive/capacitive) of various circuit admittances.The circuits have been analyzed in detail taking into account the various non-ideal parameters of the OTRA and the workability of the derived specific inverse filters has been confirmed by SPICE simulations using an exemplary CMOS OTRA architecture implemented in 0.18 µm CMOS technology, as well as by OTRAs implemented with AD844 ICs.Centre/cut-off frequencies of the order of 1.59 MHz in case of the former and of the order of 159 kHz in the case of latter have been found to be realisable.

Fig. 5 :
Fig. 5: Frequency Response of the Inverse Low Pass Filter.
Tab. 1: Comparison with previously known circuits.Model parameters of NMOS and PMOS transistors.