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      <text>23.1.2.5 Nonsinusoidal periodic functions</text>
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      <text><![CDATA[<h4>Nonsinusoidal periodic functions</h4>
The following are examples of non-sinusoidal periodic functions (they are often called "waves").
<p></p>
<b>Square wave</b><p style="text-align: center;"><b><img src="@@PLUGINFILE@@/23-1-2-fig-5-square-wave.svg" alt="square wave" width="291" height="86" class="img-responsive atto_image_button_text-bottom"><br></b></p>

Analytically we can describe this function as follows:<p>&nbsp;
\[ 
\begin{align} 
f(t) = 
    \begin{cases}
   -1 &amp;-\pi &lt; t &lt; 0 \\
   +1 &amp;0 &lt; t &lt; \pi
   \end{cases}
\end{align} 
\text{(which gives the definition over one period)} \]</p>
\[ f(t + 2\pi) = f(t) \hspace{1cm} \text{(which tells us that the function has period } 2\pi ) \]
<p></p>
<b>Saw-tooth waves</b><p style="text-align: center;"><b><img src="@@PLUGINFILE@@/23-1-2-fig-6-saw-tooth.svg" alt="saw tooth" width="280" height="111" class="img-responsive atto_image_button_text-bottom"><br></b></p>
In this case we can describe the function as follows:
\[ f(t) = 2t \hspace{1cm} 0 &lt; t &lt; 2 \hspace{1cm} f(t+2) = f(t) \]
Here the period is \(2\), the frequency is \( \dfrac{1}{2}\) and the angular frequency is \(\dfrac{2\pi}{2}  = \pi\).
<p></p>
<b>Triangular waves</b><p style="text-align: center;"><b><img src="@@PLUGINFILE@@/23-1-2-fig-7-triangular-wave.svg" alt="triangular-wave" width="414" height="114" class="img-responsive atto_image_button_text-bottom"><br></b></p>
Here we can conveniently define the function using \( -\pi &lt; t &lt; \pi\) as the "basic period":
\[ 
\begin{align} 
f(t) = 
    \begin{cases}
   -t &amp;-\pi &lt; t &lt; 0 \\
   t &amp;0 &lt; t &lt; \pi
   \end{cases}
\end{align} 
\]
or, more concisely,
\[ f(t) = \lvert t \rvert \hspace{2cm}  -\pi &lt; t &lt; \pi \]
together with the usual statement on periodicity
\[ f(t + 2\pi) = f(t).\]]]></text>
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