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

Famous quotes containing the words transfer and/or function:

    “No sociologist ... should think himself too good, even in his old age, to make tens of thousands of quite trivial computations in his head and perhaps for months at a time. One cannot with impunity try to transfer this task entirely to mechanical assistants if one wishes to figure something, even though the final result is often small indeed.”
    —Max Weber (1864–1920)

    “The uses of travel are occasional, and short; but the best fruit it finds, when it finds it, is conversation; and this is a main function of life.”
    —Ralph Waldo Emerson (1803–1882)