Infinite Impulse Response - Transfer Function Derivation

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:

\begin{align} y & = \frac{1}{a_{0}}(b_{0} x + b_{1} x + \cdots + b_{P} x \\ & - a_{1} y - a_{2} y - \cdots - a_{Q} y) \end{align}

where:

  • is the feedforward filter order
  • are the feedforward filter coefficients
  • is the feedback filter order
  • are the feedback filter coefficients
  • is the input signal
  • is the output signal.

A more condensed form of the difference equation is:

which, when rearranged, becomes:

To find the transfer function of the filter, we first take the Z-transform of each side of the above equation, where we use the time-shift property to obtain:

We define the transfer function to be:

\begin{align} H(z) & = \frac{Y(z)}{X(z)} \\ & = \frac{\sum_{i=0}^P b_{i} z^{-i}}{\sum_{j=0}^Q a_{j} z^{-j}} \end{align}

Considering that in most IIR filter designs coefficient is 1, the IIR filter transfer function takes the more traditional form:

\begin{align} H(z) & = \frac{\sum_{i=0}^P b_{i} z^{-i}}{1+\sum_{j=1}^Q a_{j} z^{-j}} \end{align}

Read more about this topic:  Infinite Impulse Response

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