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6. Filtering of Stochastic Signals
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<p class="inspiringquote">To hear is to forget. <br />
To see is to remember. <br />
To do is to learn! <br />
…Chinese proverb</p>
<h1 id="how-to-use-this-ibook">How to use this iBook<a class="headerlink" href="#how-to-use-this-ibook" title="Permanent link">¶</a></h1>
<p>This textbook is being prepared in the context of modern electronic publishing. By that we mean that modern digital technology—such as the Internet, an iPad<sup id="fnref:apple_trade"><a class="footnote-ref" href="#fn:apple_trade">1</a></sup> or Android<sup id="fnref:android_trade"><a class="footnote-ref" href="#fn:android_trade">2</a></sup> tablet, a smartphone, and a laptop or desktop computer—makes it possible to present not only text, equations, and figures but dynamic illustrations, audio samples, both internal and external links to relevant material, and interactive laboratory experiments. Extensive use of these possibilities is made in this iBook. This “user’s guide” describes how you can make use of these possibilities.</p>
<p>This edition is primarily intended for use with a browser such as Safari, Firefox, Chrome, or Opera on a laptop or desktop computer. You can, of course, use this edition through a browser on a tablet or smartphone. The 26 Laboratory Experiments in this edition involve the use of the screen, speakers, keyboard, and a graphical user interface (GUI).</p>
<p>An extended version of this iBook with an additional 32 Laboratory Experiments for a total of 58 experiments makes use of the microphone and camera that can be found on tablets and smartphones. The current implementation of this extended version is free and can be found <a href="https://apps.apple.com/us/app/stochastic-signal-processing/id1450268179?ls=1">here</a>.</p>
<h2 id="highlighting">Highlighting<a class="headerlink" href="#highlighting" title="Permanent link">¶</a></h2>
<p>Main results will be highlighted with a background in <a class="mainresult" href="">pink</a> and special cases of main results will be highlighted with a background in <a class="specialresult" href="">yellow</a>. </p>
<p class="mainresult">$$
\begin{aligned}
{S_{yy}}(\Omega ) = {\left| {H( - \Omega )}
\right|^2}{S_{xx}}(\Omega )\;
\end{aligned}
$$</p>
<p class="specialresult">$$
\begin{aligned}
{S_{yy}}(\Omega ) = {\left| {H(\Omega )}
\right|^2}{S_{xx}}(\Omega )\;
\end{aligned}
$$</p>
<p>Special cases might result, for example, when we consider only real signals or systems. In general we will consider that all signals and systems are complex, having both a real and an imaginary part. When the special case of a real signal or real system is being considered, it will be explicitly stated. </p>
<h2 id="outside-this-device">Outside this device<a class="headerlink" href="#outside-this-device" title="Permanent link">¶</a></h2>
<p>Occasionally, references will be made to other sources of information through an external <em>link</em>. Click <a href="http://www.imphys.tudelft.nl/">here</a> to see an example. When you are finished viewing the external material, return to this iBook and it should continue at the place where you left off.</p>
<h2 id="getting-around">Getting around<a class="headerlink" href="#getting-around" title="Permanent link">¶</a></h2>
<p>Endnotes are also linked within the iBook. When you see an endnote indicator<sup id="fnref:endnote"><a class="footnote-ref" href="#fn:endnote">3</a></sup>, you can click on it and be taken to the note which is at the end of the chapter. When you want to jump to the homework problems or the laboratory exercises at the end of a chapter, tap the top of the screen.</p>
<h2 id="sound-of-music">Sound of music<a class="headerlink" href="#sound-of-music" title="Permanent link">¶</a></h2>
<p>Dynamic examples, specifically music and animations, have been built into this iBook. To <em>hear</em> the results of various operations on a stochastic signal, “click” on the “playback” icon (<img src="images/play_triangle1.png" style="vertical-align:middle;">) and the sound will be played. This is a good way to test and adjust the sound level of your iBook. </p>
<table class="imgtxt" style="margin:1em auto; padding-top:10px;">
<tr>
<td style="width:auto; vertical-align:middle; text-align:center;">
<audio controls><source src="media/Paganini_Original.wav" type="audio/mpeg"></audio>
</td>
</tr>
</table>
<figcaption style="margin: 0px 30px;">
<b>Audio 1.1:</b> Click the play triangle ( <img src="images/play_triangle1.png" style="vertical-align:middle;">) to hear a brief selection of Paganini.
</figcaption>
<h2 id="at-the-movies">At the movies<a class="headerlink" href="#at-the-movies" title="Permanent link">¶</a></h2>
<p>Animations and movies will be used to illustrate various aspects of stochastic signal processing. In laboratory exercises, for example, you may need to whistle. This famous film clip is taken from the wonderful 1944 Warner Brothers film <em>“To Have and Have Not”</em> which starred Humphrey Bogart and Lauren Bacall. This clip is used under the <em>fair use</em> copyright policy<sup id="fnref:us_copyright_act"><a class="footnote-ref" href="#fn:us_copyright_act">4</a></sup>. </p>
<table class="imgtxt" style="margin:1em auto;">
<tr>
<td style="width:auto; vertical-align:middle; text-align:center;">
<video id="theVideo" src="media/Just Whistle.m4v"
poster="media/Just_Whistle-poster.jpg" width=100% controls
onended="rewind()" style="border: 2px solid rgb(174,24,16)"
playsinline webkit-playsinline></video>
</td>
</tr>
</table>
<figcaption style="margin: 0px 30px;">
<b>Movie 1.1:</b> Click the play triangle ( <img src="images/play_triangle1.png" style="vertical-align:middle;">) to learn how to whistle.
</figcaption>
<h2 id="enhanced-experience">Enhanced experience<a class="headerlink" href="#enhanced-experience" title="Permanent link">¶</a></h2>
<p>And to prevent the reader from falling into an etymological lacuna, it is possible to “point” to a word and then be offered a definition through a dictionary.</p>
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## Laboratory Exercises {: .labexp }
### Laboratory Exercise 1.1
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<p>iPad is a trademark of Apple Inc., registered in the U.S. and other countries. <a class="footnote-backref" href="#fnref:apple_trade" title="Jump back to footnote 1 in the text">↩</a></p>
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