## IIR Design Edit

### Impulse Invariant Transform Edit

The transform is

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

### Bilinear Transform Edit

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

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 and )

To go from an analogue angular frequency to a digital frequency , use

The inverse, from digital to analogue is then

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 ) are then for the centre frequency and for the bandwidth.

Using the second conversion formula, we get

rad/sec

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.