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:

    “I have proceeded ... to prevent the lapse from ... the point of blending between wakefulness and sleep.... Not ... that I can render the point more than a point—but that I can startle myself ... into wakefulness—and thus transfer the point ... into the realm of Memory—convey its impressions,... to a situation where ... I can survey them with the eye of analysis.”
    —Edgar Allan Poe (1809–1849)

    “If the children and youth of a nation are afforded opportunity to develop their capacities to the fullest, if they are given the knowledge to understand the world and the wisdom to change it, then the prospects for the future are bright. In contrast, a society which neglects its children, however well it may function in other respects, risks eventual disorganization and demise.”
    —Urie Bronfenbrenner (b. 1917)