## IIR Design Edit

### Impulse Invariant Transform Edit

The transform is

$ \frac{1}{s+b} \to \frac{1}{1-e^{-bT}z^{-1}} $

You generally need to split the analogue transfer function up into partial fractions so that you can use the above.

#### A Useful Partial Fractions Identity To Do That Edit

$ \frac{k}{(s-s_1)(s-s_2)} \equiv \frac{k}{s_1-s_2}\left(\frac{1}{s-s_1} - \frac{1}{s-s_2}\right) $

### Bilinear Transform Edit

Yields stable digital filters from stable analogue filters. The main transform is

$ s \to \frac{2}{T}\frac{1-z^{-1}}{1+z^{-1}} $

The Bilinear Transform maps the *entire* s-plane imaginary axis onto the z-plane unit circle, without any aliasing. This is good.

**However**, the Bilinear Transform also warps the frequency axis in a nonlinear way. This is bad. The specifications of the desired digital filter must be anti-warped to produce the specifications of the analogue filter to be transformed.

#### How to Pre-warp Your Filter Specs Edit

(While reading this, it is useful to remember that $ \omega = 2 \pi f $ and $ \theta = \frac{2 \pi f}{f_{s}} = 2 \pi f T $)

To go from an analogue angular frequency $ \omega $ to a digital frequency $ \theta $, use

$ \theta = 2 \tan^{-1} \left( \frac{\omega T}{2} \right) $

The inverse, from digital to analogue is then

$ \omega = \frac{2}{T} \tan \left( \frac{\theta}{2} \right) $

So say you wanted a digital band-pass filter with a centre frequency of 1000Hz and a bandwidth of 150Hz, and the signal has a sampling frequency of 10KHz.

The digital frequencies (using $ \theta = \frac{2 \pi f}{f_{s}} $) are then $ \frac{\pi}{5} $ for the centre frequency and $ \frac{3\pi}{100} $ for the bandwidth.

Using the second conversion formula, we get

$ \omega_{c} = 20000 \tan \left( \frac{1}{2} \frac{\pi}{5} \right) = 6498.4 $ rad/sec

$ \omega_{b} = 20000 \tan \left( \frac{1}{2} \frac{3\pi}{100} \right) = 943.2 $ rad/sec

You may now design your analogue filter to these specs. When you do the Bilinear transform, the resulting digital filter will have the original specs.

For a second order filter, it is often easier to leave the 2/T factor unsimplified when performing pre-warping, as it will generally cancel out after the transformation.