A Switched Capacitor based Realization of

Fractional Order Low-pass Filters

A Presentation by Ranjan Das (PhD Scholar, IIT Bombay)

Background

• Conventional Digital Signal Processing (DSP) deals with rational pole-zero models of FIR/IIR structures and looks at stability, causality, and other properties of these models.

• Many physical signals and other phenomena have been shown to possess inherently fractional order dynamics and hence fractional calculus is naturally able to model these processes with greater accuracy.

• Hence, fractional order signal and system theory will play a pivotal role in the coming eras of scientific development.

Current Work: An Overview

• Realisations of a particular family of fractional order low-pass filters, all having a cut-off frequency 1 rad/s. We varied the order of the filter from 0.2 to 0.9 in steps of 0.1 .

• We rationalized those fractional order systems using Charef’s approximation technique [1].

• The approximated transfer functions are realized using operational amplifiers and passive components.

Overview …contd

• The designed circuits were tested in OrCAD pSpice platform.

• The resistances were replaced by switch capacitor approximations, and then these modified circuits were simulated in the same platform.

• To verify correctness, the result were compared with ideal responses.

Charef’s Approximation Technique

The transfer function we started with was,

(m= fractional order < 1)

By Charef’s method, this

function will be approximated as-

Following formulae were used to get the values of zi and pi

The Poles and Zeros as Computed From the Charef’s Method :

Ideal Simulation for the Generated Transfer Functions in MATLAB:

Final Circuit with Stray-Insensitive Configuration:

Final Simulation Results

Comparison with Ideal Responses

Order Ideal Drop-rate Observed drop-rate at Corner Frequencies

0.3 6 dB/decade 6.93 dB/decade

0.4 8 dB/decade 9.54 dB/decade

0.5 10 dB/decade 11.4 dB/decade

0.6 12 dB/decade 12.04 dB/decade

0.7 14 dB/decade 14.52 dB/decade

0.8 16 dB/decade 16.12 dB/decade

0.9 18 dB/decade 18.58 dB/decade

Conclusion

References: