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## 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.