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  {
   "cell_type": "markdown",
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   "source": [
    "# Apply a frequency response function\n",
    "\n",
    "Objects from `thztools`: `apply_frf`, `timebase`, `wave`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "from matplotlib import pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "import thztools as thz"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "## Set the simulation parameters"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "n = 256  # Number of samples\n",
    "dt = 0.05  # Sampling time [ps]\n",
    "a = 0.5 # Scale factor\n",
    "eta = 1.0 # Delay [ps]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulate an input waveform"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "t = thz.timebase(n, dt=dt)\n",
    "mu = thz.wave(n, dt=dt)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Define the frequency response function\n",
    "\n",
    "The frequency response function `frfun` rescales the input waveform by `a` and delays it by `eta`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def frfun(omega, _a, _eta):\n",
    "    return _a * np.exp(-1j * omega * _eta)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Apply the frequency response function\n",
    "\n",
    "Generate the time-domain output waveform `psi` by applying `frfun` to `mu` with the `apply_frf` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi = thz.apply_frf(frfun, mu, dt=dt, args=(a, eta))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Show the input and output waveforms"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "_, ax = plt.subplots()\n",
    "\n",
    "ax.plot(t, mu, label=r'$\\mu$')\n",
    "ax.plot(t, psi, label=r'$\\psi$')\n",
    "\n",
    "ax.legend()\n",
    "\n",
    "ax.set_xlabel('Time (ps)')\n",
    "ax.set_ylabel(r'Amplitude (units of $\\mu_{p})$')\n",
    "\n",
    "ax.set_xticks(np.arange(0, 11, 5))\n",
    "ax.set_xlim(0, 10)\n",
    "\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Change sign convention\n",
    "\n",
    "The above example uses the $+i\\omega t$ sign convention for representing harmonic waves with complex exponentials. NumPy also uses this convention in its definition of the FFT. Physicists commonly use the $-i\\omega t$ sign convention, which can cause confusion when translating between analytic expressions and computational expressions. The `apply_frf` function supports both sign conventions through the boolean parameter `numpy_sign_convention`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "def frfun_phys(omega, _a, _eta):\n",
    "    return _a * np.exp(1j * omega * _eta)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "psi_phys = thz.apply_frf(frfun_phys, mu, dt=dt, args=(a, eta), numpy_sign_convention=False)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "_, ax = plt.subplots()\n",
    "\n",
    "ax.plot(t, mu, label=r'$\\mu$')\n",
    "ax.plot(t, psi_phys, label=r'$\\psi$')\n",
    "\n",
    "ax.legend()\n",
    "\n",
    "ax.set_xlabel('Time (ps)')\n",
    "ax.set_ylabel(r'Amplitude (units of $\\mu_{p})$')\n",
    "\n",
    "ax.set_xticks(np.arange(0, 11, 5))\n",
    "ax.set_xlim(0, 10)\n",
    "\n",
    "plt.show()"
   ]
  }
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