A Basic Digital Filter

What happens when we add a signal to a one-sample delayed version of itself?
$\displaystyle y[n] = x[n] + x[n - 1]$

Figure 3: A simple digital filter block diagram.
$\begin{figure}\begin{center} \begin{picture}(2.7,1.2) \put(0.27,0.0){\epsfig{f... ....16){$z^{-1}$} \put(2.42,0.93){$y[n]$} \end{picture} \end{center} \end{figure}$
Consider the following input signals:
- $x[n] = \{ \ldots, 1, 1, 1, 1, \ldots \} \hspace{0.2in} (f = 0 \mbox{ or DC})$
- $x[n] = \{ \ldots, 1, -1, 1, -1, \ldots \} \hspace{0.2in} (f = f_s / 2)$
The filter's frequency response magnitude is shown in Fig. 4.

Figure 4: Magnitude frequency response of simple digital filter.
$\begin{figure}\begin{center} \epsfig{file=figures/one-zero-mag.eps, width=3.5in} \end{center} \end{figure}$

**Figure 3:** A simple digital filter block diagram.
$\begin{figure}\begin{center} \begin{picture}(2.7,1.2) \put(0.27,0.0){\epsfig{f... ....16){$z^{-1}$} \put(2.42,0.93){$y[n]$} \end{picture} \end{center} \end{figure}$

**Figure 4:** Magnitude frequency response of simple digital filter.
$\begin{figure}\begin{center} \epsfig{file=figures/one-zero-mag.eps, width=3.5in} \end{center} \end{figure}$

A partial C / C++ implementation:

double lastInput = 0;

double Filter::tick( double input )
{
  double output = input + lastInput;   // do filter calculation
  lastInput = input;                   // save the filter state

  return output;                       // output the value
}

The filter can be implemented in Pd as shown below:
The Pd delay~ object implements a delay, with arguments specified in samples. The first argument is the maximum delay length and the second argument is the initial delay (default = 0). The delay~ object cannot be used in a feedback loop.