.. _FourierForLoop:

Fourier For Loop
========================================

For this assignment, we will both take a sneak peek at using arrays in
MATLAB and we will also practice writing a simple ``for`` loop.

First, enter the following commands into the MATLAB Command Window.
In these lines of code, you can see that the variables ``t`` and ``y`` are
arrays (also called vectors). You can also see that MATLAB has the ability
to operate at the array level, which eliminates the need to write a ``for``
loop to assign each value of the ``y`` array.

::

    >> T = 5;
    >> t = 0:15/100:15;
    >> y = sin(2*pi*t/T);
    >> plot(t,y);

Did you see three cycles of a sine wave?

Next we create a sum of sine waves, which is called 
a Fourier series. A square wave with a period of
:math:`T` can be expressed with the following Fourier series.

.. math::

    y(t) = \frac{4}{\pi} \sum_{n=1,3,5 \ldots}^{\infty} 
        \frac{1}{n} \, \sin( n\,2\,\pi\,t / T)

.. seealso:: `Fourier Series Wikipedia page
              <https://en.wikipedia.org/wiki/Fourier_series>`_.

Below are instructions to create the Fourier Series of a square wave
in MATLAB.  **Complete the assignment on MATLAB Grader:**
`MATLAB Grader Fourier For Loop Assignment
<https://grader.mathworks.com/courses/115347-applied-data-analysis-and-tools-fall-2024/assignments/315702-fourier-for-loop>`_.

#. Create a new script.
   
#. Copy the code for creating the variables ``T`` and ``t`` into the script. 

#. Add the line: ``y = zeros(1,101);``. This lines gives the ``y`` array
   initial values of zero.

#. Write a ``for`` loop where a variable called ``n`` has the values of 1,
   3, 5, and 7 during the successive iterations of the loop. 

#. In the code block of the ``for`` loop enter the Fourier Series sum from
   above. **HINT:** Add a new sine term to the existing ``y`` array each
   time through the loop.

#. After the loop code, multiply ``y`` by :math:`4/\pi`.

#. Plot the ``y`` array against ``t``. What could you change in your code to
   make the plot look more like a square wave?

.. raw:: latex

    \clearpage