BIEN425 – Lecture 14 By the end of the lecture, you should be able to: Design and implement IIR filters using frequency transform and bilinear transform Compare the advantages and disadvantages of IIR filter design strategies (zero-pole versus freq-transform and bilinear transform)

In general This is an alternative way of representing Method 2 from the last lecture. This time, we don’t even need to do partial fraction expansion, the variable in s-domain is simply changed into z-domain.

Prove:

Bilinear transformation Simply going between s-plane and z-plane This is very fun…. A circle becomes a rectangle and a line becomes a arc. Lecture14.m

Frequency warping Given bilinear transformation and s = j2pF Let’s look at how freq in analog filters (F) can be translated to freq in digital filters (f) Eq 8.83

Learning through example Building a digital lowpass filter from Chebyshev-I given our digital specs: (Butterworth example 8.8) f0 = 2.5Hz, f1 = 7.5Hz dp = 0.1, ds = 0.1 (Could have given Ap and As instead) Recall the following procedure: Exam 8.8

Step 1a) pre-wrap frequencies to analog specs Step 1b) compute r, d, minimum order

Step 1c) Find poles Step 1d) Write H(s)

Step 2) Determine fs =20hz Step 3) Re-write into H(z)

Another example Given Find the resonance frequency of this filter

Analog frequency transformation Design digital HP,BP,BS filters Always start off with a normalized lowpass filter Normalized Lowpass Filter (Analog) Analog Frequency Transformation (Analog) Bilinear Transformation (Digital)

Digital frequency transformation Normalized Lowpass Filter (Analog) Bilinear Transformation (Digital) Digital Frequency Transformation (Digital)

Beware of potential problems Double check your filter response after design Stability? Actual frequency response Impulse response (check for limit cycles or deadband effects: oscillations even when input has gone to zero)