IIR Filter Design: Basic Approaches Most common approach to IIR filter design: (1)Convert specifications for the digital filter into equivalent specifications for an analog prototype lowpass filter (2)Determine the analog lowpass filter transfer function (3) Transform into the desired digital transfer function

Digital Filter Design: Basic Approaches An analog transfer function to be denoted as where the subscript “ a ” specifically indicates the analog domain A digital transfer function derived from will be denoted as

Digital Filter Design: Basic Approaches Basic idea behind the conversion of into is to apply a mapping from the s- domain to the z-domain so that essential properties of the analog frequency response are preserved Thus mapping function should be such that: –Imaginary axis in the s-plane be mapped onto the unit circle of the z-plane –A stable analog transfer function be mapped into a stable digital transfer function

S plane to Z plane mapping s-plane z-plane analogdigital Preserve stability: Pole in the right half plan should map inside the circle in the z plan.

Euler Approximation Is the sampling interval s-plane

- s-plane to z-plane conversion - any mapping than maps stable region is s-plane (left half plane) to stable region in z-plane (inside u.c) ? bilinear transform! or * T d inserted for convention may put to any convenient value for practical use. IIR Filter Design by Bilinear Transformation (1) Design Concept

(2) Properties

IIR Digital Filter Design: Bilinear Transformation Method Bilinear transformation Above transformation maps a single point in the s-plane to a unique point in the z-plane and vice-versa Relation between and is then given by * T inserted for convention may put to any convenient value for practical use.

Bilinear Transformation Digital filter design consists of 4 steps: (1) Develop the specifications of H D (z) (2) Develop the specifications of (3) Design (4) Determine H D (z) by applying bilinear transformation to

* IIR Filter Design Procedure Given specification in digital domain Convert it into analog filter specification Design analog filter (Butterworth, Chebyshov, elliptic):H(s) Apply bilinear transform to get H(z) out of H(s)

Design a digital filter equivalent of a 2nd order Butterworth low-pass filter with a cut-off frequency fc = 100 Hz and a sampling frequency fs = 1000 samples/sec. The normalised cut-off frequency of the digital filter is given by the following equation: the equivalent analogue filter cut-off frequency ωac, The value of K is immaterial so let K = 1.

H(s) for a Butterworth filter is: Hence the transfer function of the Butterworth filter becomes:

Next, convert the analogue filter into an equivalent digital filter by applying the bilinear z-transform. This is achieved by making a substitution for s in the transfer function. The finite difference equation of the filter is found by inverting the transfer function

Direct form 2 nd order

Direct realisation for a 2nd order Butterworth equivalent filter.

Matlab Bilinear a=1; b=[1, 1.141, 1]; [c, d]=bilinear(a, b, 1000);