.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/convolution.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_auto_examples_convolution.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_auto_examples_convolution.py:


Sympy notebook with the convolution
===================================
In this notebook we use *sympy*, a CAS for python, to calculate the convolution of an
gaussian with an one-sided decaying exponential. Since *sympy* is not an requirement of
*skultrafast*, it may be manually installed.

.. GENERATED FROM PYTHON SOURCE LINES 9-14

.. code-block:: Python

    import sympy
    from sympy import (symbols, Heaviside, exp, sqrt, oo, integrate, simplify, Eq,
                       plot, pi, init_printing, solve)
    from sympy import erfc, erf
    init_printing()

.. GENERATED FROM PYTHON SOURCE LINES 15-16

First, we need to define sympy symbols.

.. GENERATED FROM PYTHON SOURCE LINES 16-21

.. code-block:: Python


    A, t, ti = symbols('A t ti', real=True)
    tau, sigma = symbols('tau sigma', positive=True)
    step = Heaviside


.. GENERATED FROM PYTHON SOURCE LINES 22-23

Define :math:`y=A\exp(-t/\tau)` and the Gaussian IRF.

.. GENERATED FROM PYTHON SOURCE LINES 25-28

.. code-block:: Python

    y = step(t) * A * exp(-t / tau)
    y


.. GENERATED FROM PYTHON SOURCE LINES 29-32

.. code-block:: Python

    irf = 1 / sqrt(2 * pi * sigma**2) * exp(-t**2 / (2 * sigma**2))
    irf


.. GENERATED FROM PYTHON SOURCE LINES 33-34

Calculate the covolution integral.

.. GENERATED FROM PYTHON SOURCE LINES 36-40

.. code-block:: Python

    func = integrate((irf.subs(t, t - ti) * y.subs(t, ti)), (ti, -oo, oo))
    func = simplify(func)
    func


.. GENERATED FROM PYTHON SOURCE LINES 41-43

Rewrite the `erf` with the `erfc` function:
erfc, erf = special.error_functions.erfc, special.error_functions.erf

.. GENERATED FROM PYTHON SOURCE LINES 43-46

.. code-block:: Python

    func2 = func.rewrite(erfc)
    func2


.. GENERATED FROM PYTHON SOURCE LINES 47-48

Plot the result for visualization:

.. GENERATED FROM PYTHON SOURCE LINES 48-50

.. code-block:: Python

    plot(func2.subs(sigma, 0.2).subs(tau, 2).subs(A, 1), (t, -1, 10))


.. GENERATED FROM PYTHON SOURCE LINES 51-54

Normalized derivatives of a Gaussian
------------------------------------
These are used to model coherent contributions.

.. GENERATED FROM PYTHON SOURCE LINES 54-68

.. code-block:: Python


    irf, irf.diff(t, 1), irf.diff(t, 2)

    p1 = None
    for i in range(0, 3):
        f = irf.diff(t, i)
        sol = solve(f.diff(t, 1), t)
        ext = f.subs(t, sol[0])
        if not p1:
            p1 = plot((f/ext).subs(sigma, 1), (t, -4, 4), show=False)
        else:
            p1.append(plot((f/ext).subs(sigma, 1), (t, -4, 4), show=False)[0])
        print((f/ext))
    p1.show()

.. GENERATED FROM PYTHON SOURCE LINES 69-71

.. code-block:: Python

    plot(irf.diff(t).subs(sigma, 0.2).subs(tau, 2), (t, -1, 1))
    plot(irf.diff(t, 2).subs(sigma, 0.2).subs(tau, 2), (t, -1, 1))


.. _sphx_glr_download_auto_examples_convolution.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: convolution.ipynb <convolution.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: convolution.py <convolution.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: convolution.zip <convolution.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_