Editing Infinite impulse response (section)

== Transfer function derivation ==
Digital filters are often described and implemented in terms of the [[difference equation]] that defines how the output signal is related to the input signal:

:<math>
\begin{align}
 y[n] {} = & b_0 x[n] + b_1 x[n-1] + \cdots + b_P x[n-P] \\
      & {} + a_1 y[n-1] + a_2 y[n-2] + \cdots + a_Q y[n-Q]
\end{align}
</math>

where:
*<math>\ P</math> is the feedforward filter order
*<math>\ b_i</math> are the feedforward filter coefficients
*<math>\ Q</math> is the feedback filter order
*<math>\ a_i</math> are the feedback filter coefficients
*<math>\ x[n]</math> is the input signal
*<math>\ y[n]</math> is the output signal.

A more condensed form of the difference equation is:
:<math>\ y[n] = \sum_{i=0}^P b_{i}x[n-i] + \sum_{i=1}^Q a_i y[n-i]</math>

To find the [[transfer function]] of the filter, we first take the [[Z-transform]] of each side of the above equation to obtain:
:<math>\ Y(z) = X(z)\sum_{i=0}^P b_i z^{-i} + Y(z)\sum_{i=1}^Q a_i z^{-i}</math>

After rearranging:
:<math>\ Y(z) \left[1-\sum_{i=1}^Q a_i z^{-i}\right] = X(z)\sum_{i=0}^P b_i z^{-i}</math>

We then define the transfer function to be:
:<math>
H(z) = \frac{Y(z)}{X(z)} = \frac{\sum_{i=0}^P b_i z^{-i}}{1-\sum_{i=1}^Q a_i z^{-i}}
</math>

[[Image:IIRFilter2.svg|thumb|250px|alt=Simple IIR filter block diagram|An example of a block diagram of an IIR filter. The <math>z^{-1}</math> block is a unit delay.]]