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docs/source/app_waveforms.rst

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+.. _waveforms:
+
+******************
+Built-in waveforms
+******************
+
+This section provides definitions of the functions that are used to create the built-in waveforms. Example plots are shown using the parameters: amplitude of one, frequency of 1GHz, time window of 6ns, and a time step of 1.926ps.
+
+gaussian
+========
+
+A Gaussian waveform.
+
+.. math:: I = e^{-\zeta(t-\chi)^2}
+
+where :math:`I` is the current, :math:`\zeta = 2\pi^2f^2`, :math:`\chi=\frac{1}{f}` and :math:`f` is the frequency.
+
+.. figure:: images/gaussian.pdf
+
+    Example of the ``gaussian`` waveform - time domain and power spectrum.
+
+
+gaussiandot
+===========
+
+First derivative of a Gaussian waveform.
+
+.. math:: I = -2 \zeta (t-\chi) e^{-\zeta(t-\chi)^2}
+
+where :math:`I` is the current, :math:`\zeta = 2\pi^2f^2`, :math:`\chi=\frac{1}{f}` and :math:`f` is the frequency.
+
+.. figure:: images/gaussiandot.pdf
+
+    Example of the ``gaussiandot`` waveform - time domain and power spectrum.
+
+
+gaussiandotnorm
+===============
+
+Normalised first derivative of a Gaussian waveform.
+
+.. math:: I = -2 \sqrt{\frac{e}{2\zeta}} \zeta (t-\chi) e^{-\zeta(t-\chi)^2}
+
+where :math:`I` is the current, :math:`\zeta = 2\pi^2f^2`, :math:`\chi=\frac{1}{f}` and :math:`f` is the frequency.
+
+.. figure:: images/gaussiandotnorm.pdf
+
+    Example of the ``gaussiandotnorm`` waveform - time domain and power spectrum.
+
+
+gaussiandotdot
+==============
+
+Second derivative of a Gaussian waveform.
+
+.. math:: I = 2\zeta \left(2\zeta(t-\chi)^2 - 1 \right) e^{-\zeta(t-\chi)^2}
+
+where :math:`I` is the current, :math:`\zeta = 2\pi^2f^2`, :math:`\chi=\frac{1}{f}` and :math:`f` is the frequency.
+
+.. figure:: images/gaussiandotdot.pdf
+
+    Example of the ``gaussiandotdot`` waveform - time domain and power spectrum.
+
+
+gaussiandotdotnorm
+==================
+
+Normalised second derivative of a Gaussian waveform.
+
+.. math:: I = \left( 2\zeta (t-\chi)^2 - 1 \right) e^{-\zeta(t-\chi)^2}
+
+where :math:`I` is the current, :math:`\zeta = 2\pi^2f^2`, :math:`\chi=\frac{1}{f}` and :math:`f` is the frequency.
+
+.. figure:: images/gaussiandotdotnorm.pdf
+
+    Example of the ``gaussiandotdotnorm`` waveform - time domain and power spectrum.
+
+
+gaussiandotdotdot
+=================
+
+Third derivative of a Gaussian waveform.
+
+.. math:: I = \zeta^2 \left( 3(t-\chi) - 2\zeta (t-\chi)^3 \right) e^{-\zeta(t-\chi)^2}
+
+where :math:`I` is the current, :math:`\zeta = 2\pi^2f^2`, :math:`\chi=\frac{1}{f}` and :math:`f` is the frequency.
+
+.. figure:: images/gaussiandotdotdot.pdf
+
+    Example of the ``gaussiandotdotdot`` waveform - time domain and power spectrum.
+
+
+ricker
+======
+
+A Ricker (or Mexican Hat) waveform which is the negative, normalised second derivative of a Gaussian waveform.
+
+.. math:: I = - \left( 2\zeta (t-\chi)^2 -1 \right) e^{-\zeta(t-\chi)^2}
+
+where :math:`I` is the current, :math:`\zeta = 2\pi^2f^2`, :math:`\chi=\frac{1}{f}` and :math:`f` is the frequency.
+
+.. figure:: images/ricker.pdf
+
+    Example of the ``ricker`` waveform - time domain and power spectrum.
+
+
+sine
+====
+
+A single cycle of a sine waveform.
+
+.. math:: I = R\sin(2\pi ft)
+
+and
+
+.. math::
+
+    R =
+    \begin{cases}
+    1 &\text{if $ft\leq1$}, \\
+    0 &\text{if $ft>1$}.
+    \end{cases}
+
+:math:`I` is the current, :math:`t` is time and :math:`f` is the frequency.
+
+.. figure:: images/sine.pdf
+
+    Example of the ``sine`` waveform - time domain and power spectrum.
+
+
+contsine
+========
+
+A continuous sine waveform. In order to avoid introducing noise into the calculation the amplitude of the waveform is modulated for the first cycle of the sine wave (ramp excitation).
+
+.. math:: I = R\sin(2\pi ft)
+
+and
+
+.. math::
+
+    R =
+    \begin{cases}
+    R_cft &\text{if $R\leq 1$}, \\
+    1 &\text{if $R>1$}.
+    \end{cases}
+
+where :math:`I` is the current, :math:`R_c` is set to :math:`0.25`, :math:`t` is time and :math:`f` is the frequency.
+
+.. figure:: images/contsine.pdf
+
+    Example of the ``contsine`` waveform - time domain and power spectrum.
+
+