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 (technically you could get the same recursive effect in a Feed-Forward system if it was infinitely long, but practically speaking, IIR means Feedback; its kind of like loop unrolling a Recursive function and calling it non-recursive).

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

• 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
    • High order systems can be decomposed into parallel low order systems, however.
  • 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