Electronically Tunable Fractional Order All Pass Filter

In this paper, an electronically tunable fractional order all pass filter (FOAPF) based on operational transconductance amplifier (OTA) is presented. It uses two OTAs and single fractional order capacitor (FC) of non-integer order α to provide FOAPF of α order. Two different values of α, in particular 0.5 and 0.9, for FC are taken for investigation. The functionality of the proposal is verified through SPICE simulations using TSMC 0.18 μm Complementary Metal Oxide Semiconductor (CMOS) process parameters. Simulated and theoretical frequency and time domain responses are found to be in close agreement.

This work presents electronically tunable fractional order all pass filter (FOAPF). Two FOAPFs [39], [40] are available in open literature to the best of authors' knowledge but these lacks in electronic tuning feature. The remaining part of this paper is organized as follows. Section 2 is divided into three subsections which include FOE, OTA and realization of FOAPF. Section 3 deals with theoretical and simulation results and finally conclusion is placed in section 4.

Fractional Order Element (FOE)
The impedance function of an FOE (denoted as F) may be expressed as , where α represents non-integer value and is termed as fractional order. The FOE impedance function can be represented in terms of magnitude ( C a   ) and phase ( 2 /  ) functions. It may be noted that though, its magnitude function is frequency dependent but phase is independent of frequency and has a constant value for a given fractional order. Therefore, the FOE is also termed as a constant phase element (CPE). The FOE is not commercially available though, its behavior can be simulated by finding an appropriate rational approximation [42] of its impedance function. The FOE is versatile element and is used to generate various basic structures such as fractional order capacitor (FC), fractional order inductor (FI), differentiator [43], [44], integrator [43], [44] inverted-L type [42] and the derived RC and RL circuits as imaginary impedance [45] in fractional domain and is found in designing of various fractional order analog circuits.

Operational Transconductance Amplifier (OTA)
The circuit symbol of OTA is given in Fig. 1. It processes differential voltage and provides output current as   Fig. 3(a) shows FOAPF based on OTA which is generalized by replacing traditional capacitor with FC in structure [47]. The impedance function of FC gives magnitude as  

Realization of Fractional Order All Pass Filter (FOAPF)
, the frequency response of (3) becomes The magnitude and phase of FOAPF may be expressed by (5) and (6) respectively It has been seen [40] that the gain at . In this work the behavior of FC is emulated through RC ladder network as shown in Fig. 3(b) which is based on 4 th order CFE approximation.

CIRCUIT SIMULATION
The proposed FOAPF is functionally verified through SPICE simulation wherein CMOS based schematic of OTA [46] is used with 0.18 µm TSMC CMOS process parameters.  The time domain response of proposed filter is shown in Fig. 6 where inputs (i) 10 mV amplitude and 1 kHz frequency sinusoidal signal for order α = 0.5 as shown in Fig. 6

CONCLUSION
This paper proposes an electronically tunable FOAPF by generalizing first order all pass filter into fractional domain where tuning operation is executed through bias currents of OTAs. The phase shift operation is also tuned electronically. It is shown that all pass filter has more flexibility in shaping the filter response when generalizing into fractional domain. The frequency and time domain responses of FOAPF of order 0.5 and 0.9 are depicted which are close to theoretical results.