Lab week1: Complex numbers
==========================
We will plot several times using matplotlib in Python. Read section 1.2 of DSP
notes on `plotting signals 
<https://staff.fnwi.uva.nl/r.vandenboomgaard/SP20162017/SystemsSignals/plottingsignals.html>`_
carefully! You need to use many of these plotfunctions several times.
Results can be uploaded in Blackboard. Please (PLEASE!!) do NOT use zip files or tarballs.

Assignment 1.
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Finish these exercises and submit the files as 1_1.py in Blackboard.

Assignment 2.
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Plot the following complex numbers as vector using the Python library matplotlib:

- 4 + 3j
- -3 - 5j
- 6 - 2j
- -2 + 4j

Upload the source as 1_2.py in Blackboard.

Assignment 3.
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Plot the sum of the previous complex numbers. Upload the Python program as
1_3.py

Assignment 4.
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Determine the polair representation of the 4 complex numbers of assignment 1
using Python.
Sumbit as 1_4.py

Assignment 5.
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The derative of

.. math::
   \frac{d{cos(\phi) + j sin(\phi)}}{d\phi} = -sin(\phi) + j cos(\phi)

Rewritten with :math:`\frac{df(\phi)}{d\phi} = i f(\phi)`

.. math::
    \frac{d(e^{\phi})}{d\phi} = j e^{j\phi}

(Check this!)

For small :math:`\phi` :

.. math::
    e^x \approx 1 + x + \frac{x^2}{2} 

and 

.. math::
    cos(x) \approx 1 - \frac{x^2}{2}, sin(x) \approx x

Plot cos(x) and sin(x) for the values 0 - 0.5. Use about 40 points and plot
these on a red line with blue points.
Do the same for :math:`e^x`. Upload a jpg or png picture file.

Assignment 6.
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Write an expression for :math:`e^{j\phi}` using only the cos(x) and sin(x)
rules of above. Do the same for :math:`e^{-j\phi}`.
Derive an expression for :math:`cos(\phi)` and :math:`sin(\phi)` as function of
:math:`e^{\phi}`.
Upload a file 1_6.txt or pdf containing the expressions.

Assignment 7.
-------------
.. math::
    z = 5e^{3.665j}

Determine with Python x and y.

Plot this function and it it's conjugated.

Create with matplotlib a label for both vectors. Upload a png or jpg picture as
1_7.png.

Assignment 8.
-------------
 :math:`z_1 = 3 + 5j` en :math:`z_2 = 7 - 4j`

Determine :math:`\|z_1|^2` and :math:`|z_2|^2`.

What strikes when you take the conjugated of z1 and z2 and multiplies
with z1, respectively z2?

- Determine :math:`z = z_1 + z_2` and :math:`z = z_1 - z_2`
- Determine :math:`z = z_1z_2`
- Determine :math:`z = z_1/z_2` and :math:`z = z_2/z_1`

What strikes when you compare the sum and difference with results of the multiplication and division?

Upload the results in blackboard as 1_8.pdf (or txt).