In 1979, Adams  proposed the concept of log-domain signal processing but this concept did not receive much attention of the researchers at that time. The power of log-domain technique came into popular focus only when Frey   gave a generalized method to synthesize log-domain filters by using state-space technique. The principle of log-domain signal processing is to first compress (logarithmic) the input signal and then process it and finally expand (exponential) the signal at output stage. The working nature of log-domain filters is the same as that of companding circuits which were proposed by Tsividis, Gopinathan and Toth  independently in 1990. Thus, the log-domain circuits fall into the class of externally linear and internally nonlinear (ELIN) circuits. Adams circuit was the first ELIN circuit. The log-domain filters are also recognized as translinear (TL) filters (or dynamic translinear filters). The TL filter concept was reinvented by Seevinck  in 1990.
Initially, log-domain filters were synthesized by using the exponential nature of bipolar transistor, but in 1994 Toumazou, Ngarmnil and Lande  proposed the first log-domain filter for implementation in MOS technology in which the MOS transistors were operated in subthreshold (or weak inversion) region. The literature survey up till 2014 shows that the log-domain filters have received more attention of the researchers during more than three decades  .
The first-order filters have been extensively used in audio and video applications where circuit simplicity and power consumption are important parameters. Thus, during the last few decades, voltage-mode and current-mode first-order filter circuits have found significant place in literature. Among the voltage-mode and current-mode circuits, the latter fulfill the contemporary requirements such as low-power consumption, low-voltage operation, large dynamic range etc.; therefore, current-mode (CM) circuits have received much attention and from time to time, a number of current-mode first- order multifunction (low-pass, high-pass and all-pass) filters  -  have been reported earlier in the literature by various researchers. Current-mode multifunction filters employing only bipolar junction transistors and a single grounded capacitor have been proposed by Kircay and Cam   in 2006 and Arslanalp, Tola and Yuce   in 2011. In 2014, Kircay  again proposed a multifunction1 filter using MOS transistors and single grounded capacitor. In this circuit  , the MOS transistors have been operated in saturated region.
This paper proposes a MOS based multifunction first-order filter which is capable of realizing all possible first-order filters namely, low-pass, high-pas and all-pass from the same configuration. In the proposed circuit, the MOS transistors forming the core low- pass filter are operating in subthreshold region wherein MOS transistors have exponential characteristics. The validity of the proposed configuration has been confirmed through SPICE simulation results. The SPICE simulations show that the proposed circuit offers a performance which makes it suitable for low voltage, low power operation.
2. Proposed Multifunction Filter Circuit
The core block of the proposed circuit is a first-order low-pass filter which has been obtained by an appropriate modification of the four-MOSFETs translinear circuit used earlier as a normal product computation function  . The key concept to obtain multiple outputs from a single input signal is to subtract the low-pass signal from the input signal to get a high-pass response and then adding this high-pass output with the low-pass to get an all-pass output. In this sense, the methodology is similar to the one adopted in recent works   . This is the first circuit of its kind in log-domain using CMOS technology.
The proposed circuit offers the advantage of the MOS transistors operating in subthreshold region  . This circuit includes a number of current steering circuits, at appropriate locations for minimizing dc offset and producing correct outputs.
In Figure 1 MOSFETs M1-M2-M3-M4 along with a capacitor C constitute the basic first-order low-pass core. The transfer function of this circuit can be determined as follows. For the translinear loop comprised of M1-M2-M3-M4, we have the following equation for the close loop containing of VGS of the four-MOSFETs.
If the MOS transistors are operated in weak inversion region they would have exponential relationship between drain current and gate source voltage  of the form
where is the zero bias current, n = 1.5 is the subthreshold slope coefficient and VT = kT/q = 26 mV at room temperature is known as thermal voltage.
Now Equation (2) can be rearranged as
Therefore Equation (1) can be written as
Equation (4) can be simplified as
From Equation (5), we finally obtain
The value of can be obtained by differentiating Equation (2) and putting in Equation (6), thereby leading to
Rearranging Equation (7), we get the transfer function of the circuit as
Figure 1. The proposed log-domain first-order multifunction filter.
Equation (8) represents the transfer function of the first-order low-pass filter and can be expressed as:
where is the cutoff frequency of the low-pass filter.
Now the low-pass output is given by
whereas the other two current outputs namely, Ihp and Iap are obtained by
for which the required operations are carried out by the appropriate current mirrors and current repeaters as shown in Figure 1. The transistors M9-M10-M17, M13-M14-M8 and M5-M6-M7 are current steering circuits and the MOSFET pairs M11-M12 and M15-M16 are simple current mirrors.
Thus, the transfer functions of the low-pass, high-pass and all-pass filters realized by the proposed circuit are given by
From Equations (13)-(15), it turns out that the cutoff frequency (in case of low-pass and high-pass) and phase (in case of all-pass) can be electronically tuned by changing
the value of If since.
3. SPICE Simulations
The proposed circuit was simulated in SPICE employing TSMC 0.35 μm Level 3 CMOS process parameters  . The selected parameters were VDD = −VSS = 0.5 V, C = 3 pF, If = 70 nA, Io = 0.3 µA and Iin = 0.2 μA. The aspect ratios of the transistors were taken as shown in Table 2. From SPICE simulation, it has been verified that the condition required for weak inversion operation (of all the four MOSFETs M1-M2-M3-M4 of the basic low-pass core) i.e. VGS < VT is satisfied.
The result of SPICE simulations of the circuit of Figure 1 using TSMC 0.35 μm level 3 CMOS process parameters as given in Table 1 with aspect ratios of the MOSFETs as given in Table 2, are shown in Figures 2-5. Figure 2 shows the frequency response of
Table 1. TSMC 0.35 μm Level 3 CMOS process parameters.
Table 2. Aspect ratios.
Figure 2. SPICE generated frequency response for the circuit of Figure 1 for If = 70 nA.
Figure 3. Transient response of various circuits: (a) Transisent response of low-pass filter (f0 = 85 KHz); (b) Transient response of high-pass filter (f0 = 85 KHz); (c) Transient response of all-pass filter (f0 = 85 KHz).
Figure 4. Electronically tuning characteristics observed by varying the current If (20 nA, 45 nA, 70 nA, 95 nA and 200 nA).
low-pass, high-pass and all-pass. Figures 3(a)-(c) shows the transient response of the low-pass, high-pass and all-pass filters designed for cutoff frequency of f0 = 85 KHz by taking If = 70 nA. Figure 4 shows the electronic controllability of the cutoff frequency by change of dc bias current (If = 20 nA, 45 nA, 70 nA, 95 nA and 120 nA) and Figure 5 shows the phase response of the all-pass filter. The proposed circuit has been tested in 0.35 μm technology has also been tested in 0.18 μm technology as per the reviewer one. The results obtained where almost similar to results obtained by 0.35 μm technology. In the all-pass response the cutoff frequency has been found better.
Figure 5. Phase response of the all-pass filter.
The SPICE simulation results, thus confirm the validity of the proposed filter.
4. Concluding Remarks
Authors are thankful to “Analog Signal Processing Research Lab”, Electronics Engineering Department, Institute of Engineering and Technology, Lucknow for conducting this research. The authors are also thankful to World Bank assisted project “Technical Education Quality Improvement Program” for funding this lab.