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
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.