Infinite Impulse Response Filter

An Infinite Impulse Response Filter is a Linear Time-Invariant System Filter whose Impulse Response never becomes zero, or in other words, the duration of response to the Impulse Signal is infinite. This property is due to such filters being recursively defined, i.e. having Feedback.

$$\begin{array}{c}\mathcal{H}:{\mathbb{R}}^{\mathbb{Z}}\times {\mathbb{R}}^{\mathbb{Z}}\times \mathbb{Z}\to \mathbb{R}\\ \mathcal{H}(x,y,n)=y[n]\\ y[n]={\sum }_{k=0}^M{b}_{k}x[n-k]-{\sum }_{k=1}^N{a}_k\cdot y[n-k]\\ \end{array}$$

• Because IIR Filters are Feed-Back
  • They are is inherently iterative and thus better suited for the CPU, rather than GPU which is where FIR belongs
  • Caching becomes even more important
  • They can be Divergent, which can lead to Overflow. Thus, Stability needs to be considered

Z Transform

$$\begin{array}{rl}H(z)& =\frac{{\sum }_{k=0}^M{b}_k{z}^{-k}}{{\sum }_{k=0}^N{a}_k{z}^{-k}},a_0=1\\ & =\frac{b_0{\prod }_{k=1}^M(1-c_k{z}^{-1})}{{\prod }_{k=1}^N(1-d_k{z}^{-1})}\\ & \end{array}$$

• See Z Transform

Divergence

For a system with poles $d_k$

• Asymptotically Stable $⟺\forall k,|d_k|<1$
• Marginally Stable $⟺\forall k,|d_k|\le 1$ and all poles with $|d_k|=1$ are Algebraic Multiplicity of 1
• Unstable $⟺\exists k,|d_k|>1$ or $\exists k,|d_k|=1$ with Algebraic Multiplicity greater than 1

Impulse Response

$$h[n]=\sum _{k=1}^N{A}_k(d_k{)}^{n}u[n]$$

• See Impulse Response