Topic 5: IIR Filter Design

Main Methods for IIR Filter Design

1. Filter design using pole placement

2. IIR filter design by impulse invariance

3. IIR filter design by bilinear transform

IIR Filter design by pole placement

• Given the angle (w) and distance (r) of each pole / zero:

  • Calculate the value as z = re^jw

  • Define the transfer function H(z) using the poles and zeros

Filter design by impulse invariance

1. An analogue filter is designed to a given specification (given by a Laplace transfer function)

2. The analogue design is translated into a discrete filter with the same impulse response

   1. H_a(s) → h_a(t) → h[n] → H(z)

   2. Perform the inverse Laplace transform

   3. Sample the sequence

   4. Perform the Z Transform

3. Given H_a(s) with specific poles

   1. Expand with partial fractions

   2. Each pole is converted with e^sT , where s is the pole value and T is the sampling period

   3. Change the denominator from (s + s₀) to (1 - e^sTz^-1)

Bilinear Transform

• Converts a continuous time system into a discrete time system by mapping the s-domain to the z-domain

• Steps:

  1. Start with analogue transfer function H_a(s)

  2. Substitute the transformation formula

  3. Simplify to obtain H(z)

• Properties:

  • No aliasing (provides one-to-one mapping)

  • Stable systems remain stable

  • Widely used in digital filter design

Bilinear Transform Formula

FIR vs IIR stability

• FIR filters are always stable

• Stability can be difficult to judge for IIR filters - quantisation error can cause instability

FIR vs IIR order

FIR filters need to a higher order for a given filter specification (higher order for a steeper roll off)

FIR vs IIR phase response

FIR filters can be designed with a perfectly linear (non-distorting) phase response