Infinite Impulse Response (IIR) Filters

• Infinite Impulse Response (IIR) filters include delayed and scaled versions of the output signal that are fed back into the current output.

• IIR filters are described by the following difference equation:
  $$y[n] = b_{0} x[n] + b_{1} x[n - 1] + b_{2} x[n - 2] \ldots - a_{1} y[n - 1] - a_{2} y[n - 2] \ldots, \hspace{0.2in} n = 0, 1, 2, \ldots,$$
  where the input $x[n] \ensuremath{\stackrel{\Delta}{=}}0$ and output $y[n] \ensuremath{\stackrel{\Delta}{=}}0$ for $n < 0$.

• An IIR filter can be represented by a block diagram as shown in Fig. 4 below.

  Figure 4: A general IIR digital filter block diagram.

  

• With respect to the filter block diagram, IIR filters make use of both feed-forward and feedback terms.

• Given these feedback terms, IIR filters can become unstable based on the values of the feedback coefficients ($a_{k}$ terms).

• Because of this constructive feedback, an IIR filter can produce strong peaks, or resonances, in its frequency magnitude response.

• The feedback terms also result in a long filter impulse response. Though a stable filter's impulse response will decay toward zero over time, it technically never exactly reaches zero (ignoring issues of finite precision arithmetic).

• The order of an IIR filter is equivalent to the greater of the number of its delayed input or output terms.