Current difference buffered amplifier (CDBA) based universal inverse filter configuration is proposed. The topology can be used to synthesize inverse low-pass (ILP), inverse high-pass (IHP), inverse band-pass (IBP), inverse band-reject (IBR), and inverse all-pass filter functions with appropriate admittance choices. Workability of the proposed universal inverse filter configuration is demonstrated through PSPICE simulations for which CDBA is realized using current feedback operational amplifier (CFOA). The simulation results are found in close agreement with the theoretical results.
1. Introduction
Inverse filters are commonly used in communication [1], speech processing, audio and acoustic systems [2, 3], and instrumentation [4] to reverse the distortion of the signal incurred due to signal processing and transmission. The transfer characteristics of the system that caused the distortion should be known a priori and the inverse filter to be used should have a reciprocal transfer characteristic so as to result in an undistorted desired signal. Literature review on inverse filter suggests that numerous well-established [5] methods for digital inverse filter design do exist but analog inverse filter design remained unexplored area as is evident from the limited availability of analog inverse filter circuits/design methods [6–14] until recently. However recent research trend suggests that the area is now gaining a renewed interest.
A brief account of the complete literature on analog inverse filter is presented here. Reference [6] presents a general method for obtaining the inverse transfer function for linear dynamic systems and the inverse transfer characteristic for nonlinear resistive circuits using nullors. The realization procedures for the current-mode FTFN-based inverse filters from the voltage-mode op-amp-based RC filters are presented in [7, 8]. The procedure outlined in [7] is applicable to planar circuits only as it uses RC : CR dual transformation, whereas the method presented in [8] makes use of adjoint transformation and thus is applicable to nonplanar circuits [12]. Single FTFN based inverse filters proposed in [9–12] present inverse filters using current feedback operational amplifier (CFOA). All the circuits presented in [10, 11] provide single inverse filter function; however [12] presents a topology which can realize inverse low-pass (ILP), inverse high-pass (IHP), and inverse band-pass (IBP) filter functions by appropriate admittance choice. In [13, 14] inverse all-pass (IAP) filters have been implemented using current difference transconductance amplifier (CDTA) and current conveyors, respectively. This study reveals that no universal inverse filter configuration has been proposed in the literature so far, to the best of the authors’ knowledge. Therefore the aim of this paper is to present a current differencing buffered amplifier (CDBA) based universal inverse filter topology which realizes all five inverse filter functions, namely, ILP, IHP, IBP, IBR and IAP by appropriate admittance selection.
2. Circuit Description
Inherent wide bandwidth which is virtually independent of closed-loop gain, greater linearity, and large dynamic range are the key performance features of current mode technique [15]. The CDBA being a current processing analog building block inherits the advantages of current mode technique. In addition, it is free from parasitic capacitances [16] as its input terminals are internally grounded. Thus this active bolck is appropriate for high frequency operation. The circuit symbol of CDBA is shown in Figure 1, and the port characteristics are given by
(1)[IzVwVpVn]=[001-1100000000000][VzIwIpIn].
The proposed inverse filter configuration is shown in Figure 2. Routine analysis of the circuit of Figure 2 results in the following transfer function:
(2)V0(s)Vi(s)=N(s)D(s)=Y1Y3Y2Y4+Y3Y6-Y2Y5,
where
(3)N(s)=Y1Y3,D(s)=Y2Y4+Y3Y6-Y2Y5.
With the admittance choices of Y1=sC1+G1 and Y3=sC3+G3, the N(s) can be expressed as
(4)N(s)=s2C1C3+s(C1G3+C3G1)+G1G3.
And the appropriate admittance choices for Y2, Y4, Y5, and Y6, as shown in Table 1, would result in the required denominator functions D(s) and hence the required inverse filter responses.
Response type
Y2
Y4
Y5
Y6
ILP
G2
G4
0
0
IHP
sC2
sC4
0
0
IBP
sC2
G4
0
0
IBR
G2
G4
sC5
sC6
IAP
G2
G4
sC5
sC6
Block diagrammatic representation of CDBA.
Proposed inverse filter configuration.
Using the admittance choices given in Table 1, the ILP, IHP, and IBP response can, respectively, be expressed as
(5)V0Vi=1(G2G4)/(s2C1C3+s(C1G3+C3G1)+G1G3),V0Vi=1(s2C2C4)/(s2C1C3+s(C1G3+C3G1)+G1G3),V0Vi=1(sC2G4)/(s2C1C3+s(C1G3+C3G1)+G1G3).
For admittance choices suggested for IBR and IAP in Table 1, the D(s) can be written as
(6)D(s)=s2C3C6+s(C6G3-C5G2)+G2G5.
The resulting transfer function can be expressed as
(7)V0(s)Vi(s)=1s2C3C6-s(C5G2-C6G3)+G2G4s2C1C3+s(C1G3+C3G1)+G1G3
which represents an IBR response if C5=C6=C1 and G1G3=G2G5. The response will be IAP if C5G2-C6G3=C1G3+C3G1 which can be easily obtained by choosing a suitable value of C5. If C3=C6=C1=C, then C5=3C yields an IAP response provided G1=G2=G3.
The resonant angular frequency (ω0) and the quality factor (Q0) are given by (8) and (9), respectively, for all the responses
(8)ω0=(G1G3C1C3),(9)Q0=C1C3G1G3(C1G3+G1C3),
(whereas H_{ILP}, H_{IHP}, and H_{IBP}, the gain constants for ILP, IHP, and IBP responses, respectively, are given by
(10)HILP=G2G4G1G3,HIHP=C2C4C1C3,HIBP=C2G4(C1G3+C3G1).
3. Sensitivity Analysis
The passive sensitivities of ω0 and Q0 for the proposed configuration can be expressed as
(11)SG1ω0=SG3ω0=12,SG1ω0=SC3ω0=-12,SG1Q0=SC3Q0=12-C3G1(C1G3+C3G1),SG3Q0=SC1Q0=12-C1G3(C1G3+C3G1).
It is clearly observed from (11) that the passive sensitivities are lower than 1/2 in magnitude and hence the proposed universal inverse filter configuration may be termed as insensitive.
4. Realizing a CDBA and Associated Nonideality Analysis
For the proposed configuration, the CDBA was realized using AD844 CFOA IC as shown in Figure 3 [17]. Ideally the input resistance at the x terminal is zero and is infinite at the z terminal. From Figure 3 various currents can be calculated as
(12)Iz1=Ip,Ix2=In-Iz1,Iz2=Ix2.
Therefore the current from z terminal Iz can be calculated as
(13)Iz=-Iz2=(Ip-In).
And the output voltage Vw is given as
(14)Vw=Vz.
In analysis so far, ideal characteristics of the CFOA have been considered. However, the effect of the parasitics of the CFOA needs to be taken into consideration for performing nonideality analysis [18–21]. For this, the model of AD844 [18] which includes a finite input resistance Rx in series with Cx at port-x, the z-port parasitic impedance (Rz∥Cz), and the y-port parasitic impedance (Ry∥Cy) is used. Using this nonideal model for CFOA, the CDBA structure of Figure 3 modifies to Figure 4. The nonideal transfer function of ILP from Figure 4 can be expressed as
(15)V0Vi=(s2C1C3+s(C1G3+C3G1)+G1G3)/G2′G4′(1+sCz1G2′)(1+sCz2G4′)×(1+sGx1((Cx1+C1)+G1))×(1+sGx2((Cx2+C3)+G3)),
where G2′=1/(R2∥Rz1) and G4′=1/(R4∥Rz2).
CDBA realization with AD844 [17].
CDBA realization with nonideal model of AD844.
Considering Gx1≫G2 and Gx2≫G5, (15) modifies to
(16)V0Vi=(s2C1C3+s(C1G3+C3G1)+G1G3)/G2′G4′(1+sCz1G2′)(1+sCz2G4′)×(1+s(Gx1+C1)Gx1)(1+s(Gx2+C3)Gx2).
It is clear from (16) that nonidealities of CFOA introduce parasitic poles in the transfer function. The deviation from the ideal behavior so caused can be kept small if all the external resistors are chosen to be much larger than Rx but much smaller than Ry and Rz. Similarly external capacitors should be chosen to be much larger than Cy and Cz. Nonideal transfer functions for IHP and IBP can also be deduced in a similar manner.
5. Simulation Results
The proposed theoretical predictions are validated through simulations using PSPICE macromodel of CFOA AD844 IC as shown in Figure 3. Supply voltages used are ±10V. The proposed inverse filter configuration is designed with equal value components. All the resistances of value 10 KΩ and capacitors of value 50 pF are chosen resulting in a theoretical f0 of 318.5 KHz. Simulated frequency magnitude responses for ILP, IHP, IBP, IBR, and IAP are shown from Figure 5(a) to Figure 5(e), respectively, whereas Figure 5(f) shows the phase response for IAP. The simulated f0 for all the responses is found to be 316.3 KHz and is in close agreement to the theoretical value.
A current difference buffered amplifier (CDBA) based universal inverse filter configuration is proposed. Appropriate admittance selections allow using the proposed topology as one of the five inverse filter configurations, namely, ILP, IHP, IBP, IBR, and IAP. Workability of the proposed universal inverse filter configuration is demonstrated through PSPICE simulations for which CDBA is realized using current feedback operational amplifier (CFOA). The simulation results are found in close agreement with the theoretical results.
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