<img src="https://upload.wikimedia.org/wikipedia/commons/e/ea/From_Continuous_To_Discrete_Fourier_Transform.gif">
<p>[<a href="https://commons.wikimedia.org/wiki/File:From_Continuous_To_Discrete_Fourier_Transform.gif">Image1</a>]</p>
<h2>Introduction</h2>
<p>
Hey it's a me again <a href="https://peakd.com/@drifter1">@drifter1</a>!
</p>
<p>
Today we continue with my mathematics series about <strong>Signals and Systems</strong> in order to cover <strong>Sinusoidal and Complex Exponential Signals</strong>.
</p>
<p>
So, without further ado, let's get straight into it!
</p>
<hr>
<h2>Sinusoidal Signals</h2>
<p>
<img src="https://upload.wikimedia.org/wikipedia/commons/thumb/8/84/Sine_wave_amplitude.svg/679px-Sine_wave_amplitude.svg.png"><br><br>
[<a href="https://commons.wikimedia.org/wiki/File:Sine_wave_amplitude.svg">Image 2</a>]
</p>
<p>
Sinusoidal signals are periodic signals, which are based on the trigonometric functions sine (sin) and cosine (cos).
</p>
<h3>Continuous-Time Sinusoidals</h3>
<p>
In continuous-time, the generic form of a sinusoidal signal is:<br><br>
<img src="https://quicklatex.com/cache3/cf/ql_31a8d4b391f52a55f2379d65fe76ebcf_l3.png"><br><br>
where:
<ul>
  <li><em>A</em>: amplitude</li>
  <li><em>ω<sub>o</sub></em>: angular frequency</li>
  <li><em>φ</em>: phase shift</li>
</ul>
</p>
<h4>Period</h4>
<p>
The period of a sinusoidal signal is given by:<br><br>
<img src="https://quicklatex.com/cache3/2f/ql_87d4a8b7c82c7e2c3418f4773a40f82f_l3.png">
</p>
<h4>Present Ampltiude</h4>
<p>
The signal's amplitude at present (<em>time = t<sub>o</sub> = 0</em>) can be easily calculated using:<br><br>
<img src="https://quicklatex.com/cache3/68/ql_35566ec9b875abd0af9c8ae99332a168_l3.png"><br><br>
because only the phase shift <em>φ</em> affects the present value.
</p>
<h4>Time Shift - Phase Shift Relationship</h4>
<p>
Time shifting a sinusoidal signal is related to phase shifting as follows:<br><br>
<img src="https://quicklatex.com/cache3/61/ql_3955ba337d5c94c3e0c70705defdcd61_l3.png"><br><br>
Thus, in this example, the time shift by <em>t<sub>o</sub></em> is equal to a phase shift by <em>φ = ω<sub>o</sub>t<sub>o</sub></em>.
</p>
<p>
Or, the other way around, a phase shift by any <em>φ</em> implies a time shift by some unknown multiple of <em>ω<sub>o</sub></em>.
</p>
<h4>Even Sinusoidal</h4>
<p>
When the phase shift is <em>φ = 0</em>, the sinusoidal signal <em>A cos ω<sub>o</sub>t</em> is falling into the even category.
</p>
<p>
For a signal to be even the following must be true: <em>x(t) = x(-t)</em>, which is of course true for the cosine function, as:<br><br>
<img src="https://quicklatex.com/cache3/31/ql_b636dd3439956a4b134fe21211236931_l3.png">
</p>
<h4>Odd Sinusoidal</h4>
<p>
In a similar way, its also possible to prove that the sine function is odd, because:<br><br>
<img src="https://quicklatex.com/cache3/53/ql_865db3f982ab01b5bc849f1b706f8053_l3.png"><br><br>
Thus, the signal <em>A sin ω<sub>o</sub>t</em> is considered an odd signal.
</p>
<p>
Its worth noting that the sine and cosine functions/signals differ by <em>φ = -π/2</em>.
So, its easy to conclude that phase shifting by <em>φ = -π/2</em>, the cosine signal is also considered an odd signal:<br><br>
<img src="https://quicklatex.com/cache3/41/ql_f0f347aa4b83396564c788b40b29f841_l3.png">
</p>
<h3>Discrete-Time Sinusoidals</h3>
<p>
In discrete-time, the sinusoidal signal is given by:<br><br>
<img src="https://quicklatex.com/cache3/55/ql_ae7f291364ec9e438407ac9dc7000255_l3.png"><br><br>
where:
<ul>
  <li><em>A</em>: amplitude</li>
  <li><em>Ω<sub>o</sub></em>: angular frequency</li>
  <li><em>φ</em>: phase shift</li>
</ul>
</p>
<h4>Time Shift - Phase Shift Relationship</h4>
<p>
In the case of discrete-time, time shifting again implies a phase shift:<br><br>
<img src="https://quicklatex.com/cache3/e0/ql_91c3b30bb29b35f11dee548831dfdce0_l3.png"><br><br>
So, a time shift by <em>n<sub>o</sub></em> samples is equal to a phase shift by <em>Ω<sub>o</sub>n<sub>o</sub></em>.
</p>
<p>
Let's note that phase shifting now doesn't imply a time shift, as the sample rate affects the outcome of phase shifting.
As a result:<br><br>
<img src="https://quicklatex.com/cache3/55/ql_5fac8ad5692f499a876b5d3d3200c955_l3.png">
</p>
<h4>Even-Odd Sinusoidal</h4>
<p>
Similar to continuous-time, the cosine signal is again considered an even signal, whilst the sine signal an odd signal:<br><br>
<img src="https://quicklatex.com/cache3/08/ql_c0571dd5c921abc0f5817e538fa5ca08_l3.png">
</p>
<h4>Requirements for Periodicity</h4>
<p>
Any sinusoidal signal is considered an periodic signal in continuous-time, but in discrete-time things change slightly.
</p>
<p>
In general, in discrete-time a signal is considered periodic only when their exists a small integer <em>N</em> for which:<br><br>
<img src="https://quicklatex.com/cache3/d4/ql_f521e522562d5801bd97dff1984139d4_l3.png">
</p>
<p>
In discrete-time, a sinusoidal signal:<br><br>
<img src="https://quicklatex.com/cache3/a6/ql_13d0a7429d15e496f2a31a998ca214a6_l3.png"><br><br>
is periodic when <em>Ω<sub>o</sub>N</em> is a multiple of <em>2π</em>, and so the following is true:<br><br>
<img src="https://quicklatex.com/cache3/2b/ql_916e1700544a40957a71d4ea21a6e62b_l3.png"><br><br>
The period <em>N</em> is the smallest natural number for which this equation is true.
If none exists then the sinusoidal is aperiodic.
</p>
<p>
The following is a visualization of how the sample-rate affects the periodicity of a sinusoidal signal:<br><br>
<img src="https://upload.wikimedia.org/wikipedia/commons/c/cf/Aliasing_sinusoidal.gif"><br><br>
[<a href="https://commons.wikimedia.org/wiki/File:Aliasing_sinusoidal.gif">Image 3</a>]
</p>
<h3>Sinusoidals in Continuous- and Discrete-time at Distinct Frequencies</h3>
<p>
In addition to the issue of periodicity, continuous- and discrete-time sinusoidal also differ in other aspects.
</p>
<p>
In continuous-time, distinct values of the frequency <em>ω<sub>o</sub></em> result into completely distinct signals.
However, in discrete-time, values of <em>Ω<sub>o</sub></em> which are separated by <em>2π</em> result into identical signals.
</p>
<p>
So, in continuous-time, if <em>ω<sub>2</sub> ≠ ω<sub>1</sub></em> then <em>x<sub>2</sub>(t) ≠ x<sub>1</sub>(t)</em>.
But, in discrete-time, if <em>Ω<sub>2</sub> = Ω<sub>1</sub> + 2πm</em> then <em>x<sub>2</sub>[n] ≠ x<sub>1</sub>[n]</em>.
</p>
<hr>
<h2>Exponential Signals</h2>
<p>
Exponential signals can be defined as:<br><br>
<img src="https://quicklatex.com/cache3/8c/ql_0f552f91a01bbcb4437b86965ea0b28c_l3.png"><br><br>
where both <em>C</em> and <em>a</em> are real numbers.
</p>
<h3>Time Shift - Scale Change Relationship</h3>
<p>
Time shifting an exponential signal implies scale change, as follows:<br><br>
<img src="https://quicklatex.com/cache3/12/ql_d948e77e55afaa03898c036f743d3f12_l3.png">
</p>
<h3>Complex Exponentials</h3>
<p>
Replacing <em>C</em> and <em>a</em> with complex numbers results in complex explonentials, which can be easily related to sinusoidal signals using Euler's relation.
</p>
<h4>Continuous-Time</h4>
<p>
In continuous-time, an complex exponential is defined as:<br><br>
<img src="https://quicklatex.com/cache3/ec/ql_36da4208b73a42fdb6dce1f0b6a42eec_l3.png"><br><br>
where <em>C</em> and <em>a</em> tend to be defined as:<br><br>
<img src="https://quicklatex.com/cache3/3c/ql_cb5e37c50e673ba91acabedf9cfd123c_l3.png"><br><br>
which results in the following representation for complex exponentials:<br><br>
<img src="https://quicklatex.com/cache3/f0/ql_fa5cd0167fa252d7976b15f101de8ef0_l3.png">
</p>
<p>
Euler's relation allows us to replace the second exponential with a sum of cosine and sine:<br><br>
<img src="https://quicklatex.com/cache3/40/ql_b561c47b27e115985dea58a370302240_l3.png"><br><br>
And so, the final representation of complex exponentials is:<br><br>
<img src="https://quicklatex.com/cache3/95/ql_c3466a083bfb195f97fb5ccb34f93d95_l3.png"><br><br>
where the real and imaginary parts are clearly separated.
</p>
<p>
Complex exponentials can be thought of as exponentially growing or decaying sinusoidal signals, as shown below.<br><br>
<img src="https://ars.els-cdn.com/content/image/3-s2.0-B9780123747167000041-f01-05-9780123747167.jpg?_"><br><br>
[<a href="https://www.sciencedirect.com/topics/computer-science/complex-exponential">Image 4</a>]
</p>
<h4>Discrete-Time</h4>
<p>
In discrete-time, an complex exponential is defined as:<br><br>
<img src="https://quicklatex.com/cache3/f9/ql_e07eca7b94c37df3ebf0f11259f6c4f9_l3.png"><br><br>
where <em>C</em> and <em>a</em> are defined as:<br><br>
<img src="https://quicklatex.com/cache3/24/ql_77ce0bd971e70dcc2a92fe9538193324_l3.png"><br><br>
therefore resulting into the following representation:<br><br>
<img src="https://quicklatex.com/cache3/e0/ql_3199f350a45677fe5f0545312c5abbe0_l3.png"><br><br>
Using Euler's relation, the last exponential can be replaced by sinusoidal signals, giving us the following, now final, form:<br><br>
<img src="https://quicklatex.com/cache3/34/ql_a10e4ea4a614fb5beefbab81b985d234_l3.png">
</p>
<hr>
<h2>RESOURCES:</h2>
<h3>References</h3>
<ol>
  <li><a href="https://ocw.mit.edu/resources/res-6-007-signals-and-systems-spring-2011">Alan Oppenheim. RES.6-007 Signals and Systems. Spring 2011. Massachusetts Institute of Technology: MIT OpenCourseWare, License: Creative Commons BY-NC-SA.</a></li>
  <li><a href="https://www.tutorialspoint.com/signals_and_systems/">https://www.tutorialspoint.com/signals_and_systems/</a></li>
</ol>
<h3>Images</h3>
<ol>
  <li><a href="https://commons.wikimedia.org/wiki/File:From_Continuous_To_Discrete_Fourier_Transform.gif">https://commons.wikimedia.org/wiki/File:From_Continuous_To_Discrete_Fourier_Transform.gif</a></li>
  <li><a href="https://commons.wikimedia.org/wiki/File:Sine_wave_amplitude.svg">https://commons.wikimedia.org/wiki/File:Sine_wave_amplitude.svg</a></li>
  <li><a href="https://commons.wikimedia.org/wiki/File:Aliasing_sinusoidal.gif">https://commons.wikimedia.org/wiki/File:Aliasing_sinusoidal.gif</a></li>
  <li><a href="https://www.sciencedirect.com/topics/computer-science/complex-exponential">https://www.sciencedirect.com/topics/computer-science/complex-exponential</a></li>
</ol>
<p>Mathematical equations used in this article, where made using <a href="http://quicklatex.com/">quicklatex</a>.</p>
<hr>
<h2>Previous articles of the series</h2>
<ul>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-introduction">Introduction</a> &rarr; Signals, Systems</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-signal-basics">Signal Basics</a> &rarr; Signal Categorization, Basic Signal Types</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-signal-operations-with-examples">Signal Operations with Examples</a> &rarr; Amplitude and Time Operations, Examples</li>
  <li><a href="https://peakd.com/hive-196387/@drifter1/mathematics-signals-and-systems-system-classification-with-examples">System Classification with Examples</a> &rarr; System Classifications and Properties, Examples</li>
</ul>
<hr>
<h2>Final words | Next up</h2>
<p>And this is actually it for today's post! Till next time!</p>
<p>See Ya!</p>
<p><img src="https://steemitimages.com/0x0/https://media.giphy.com/media/ybITzMzIyabIs/giphy.gif" width="500" height="333"/></p>
<p>Keep on drifting!</p>