A Basic Digital Filter with Feedback

What happens if we use a previous filter output value to produce the filter's current output?
$\displaystyle y[n] = x[n] + y[n - 1]$

Figure 6: A simple digital filter block diagram.
$\begin{figure}\begin{center} \begin{picture}(2.5,1.2) \put(0.27,0.0){\epsfig{f... ...16){$z^{-1}$} \put (2.42,0.93){$y[n]$} \end{picture} \end{center} \end{figure}$
Consider the following input signals:
- $x[n] = \{ \ldots, 1, 1, 1, 1, \ldots \} \hspace{0.2in} (f = 0 \mbox{ or DC})$
- $x[n] = \{ \ldots, 1, -1, 1, -1, \ldots \} \hspace{0.2in} (f = f_s / 2)$
This system is a digital integrator.
The filter's frequency response magnitude is shown in Fig. 7.

Figure 7: Magnitude frequency response of simple feedback digital filter.
$\begin{figure}\begin{center} \epsfig{file=figures/one-pole-mag.eps, width=3.5in} \end{center} \end{figure}$
This filter is unstable, which is evident because the gain for a DC signal is infinite. When working with feedback, we must be careful to guarantee system stability.
Let's add a scalar coefficient to the system, as follows:
$\displaystyle y[n] = x[n] + a_1 * y[n - 1]$

Figure 8: A simple digital filter block diagram.
$\begin{figure}\begin{center} \begin{picture}(2.5,1.0) \put(0.27,0.0){\epsfig{f... ...6,0.26){$a_1$} \put(2.42,0.79){$y[n]$} \end{picture} \end{center} \end{figure}$
The new filter's frequency response magnitude, with , is shown in Fig. 9.

Figure 9: Magnitude frequency response of “leaky integrator” filter.
$\begin{figure}\begin{center} \epsfig{file=figures/leaky.eps, width=3.5in} \end{center} \end{figure}$
This system, referred to as a “leaky integrator”, is stable at all frequencies as long as the feedback coefficient $\vert a_1\vert < 1$ .

**Figure 6:** A simple digital filter block diagram.
$\begin{figure}\begin{center} \begin{picture}(2.5,1.2) \put(0.27,0.0){\epsfig{f... ...16){$z^{-1}$} \put (2.42,0.93){$y[n]$} \end{picture} \end{center} \end{figure}$

**Figure 7:** Magnitude frequency response of simple feedback digital filter.
$\begin{figure}\begin{center} \epsfig{file=figures/one-pole-mag.eps, width=3.5in} \end{center} \end{figure}$

**Figure 8:** A simple digital filter block diagram.
$\begin{figure}\begin{center} \begin{picture}(2.5,1.0) \put(0.27,0.0){\epsfig{f... ...6,0.26){$a_1$} \put(2.42,0.79){$y[n]$} \end{picture} \end{center} \end{figure}$

**Figure 9:** Magnitude frequency response of “leaky integrator” filter.
$\begin{figure}\begin{center} \epsfig{file=figures/leaky.eps, width=3.5in} \end{center} \end{figure}$

A partial C / C++ implementation:

double lastOutput = 0;

double Filter::tick( double input )
{
  return lastOutput = input + a1 * lastOutput;   // do filter calculation
}

The Pd delay~ object cannot be used in a feedback loop. But this filter can be implemented with the fexpr~ object as shown below: