{
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   "cell_type": "markdown",
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   "source": [
    "# Characterization of Systems in the Time Domain\n",
    "\n",
    "*This Jupyter notebook is part of a [collection of notebooks](../index.ipynb) in the bachelors module Signals and Systems, Communications Engineering, Universität Rostock. Please direct questions and suggestions to [Sascha.Spors@uni-rostock.de](mailto:Sascha.Spors@uni-rostock.de).*"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Eigenfunctions\n",
    "\n",
    "An [eigenfunction](https://en.wikipedia.org/wiki/Eigenfunction) of a system is defined as the input signal $x(t)$ which produces the output signal $y(t) = \\mathcal{H}\\{ x(t) \\} = \\lambda \\cdot x(t)$ with $\\lambda \\in \\mathbb{C}$. The weight $\\lambda$ associated with $x(t)$ is known as scalar eigenvalue of the system. Hence, besides a weighting factor, an eigenfunction is not modified by passing through the system.\n",
    "\n",
    "[Complex exponential signals](../continuous_signals/standard_signals.ipynb#Complex-Exponential-Signal) $e^{s t}$ with $s \\in \\mathbb{C}$ are eigenfunctions of linear time-invariant (LTI) systems. This can be proven by applying the properties of LTI systems. Lets assume a generic LTI system with input signal $x(t) = e^{s t}$ and output signal $y(t) = \\mathcal{H}\\{ x(t) \\}$. The response of the LTI system to the shifted input signal $x(t-\\tau) = e^{s (t-\\tau)}$ reads\n",
    "\n",
    "\\begin{equation}\n",
    "y(t - \\tau) = \\mathcal{H}\\{ x(t-\\tau) \\} = \\mathcal{H}\\{ e^{-s \\tau} \\cdot e^{s t}  \\}\n",
    "\\end{equation}\n",
    "\n",
    "due to the implied shift-invariance. Now considering the implied linearity this can be reformulated as\n",
    "\n",
    "\\begin{equation}\n",
    "y(t - \\tau) =  e^{-s \\tau} \\cdot \\mathcal{H}\\{ e^{s t}  \\} = e^{-s \\tau} \\cdot y(t)\n",
    "\\end{equation}\n",
    "\n",
    "It is straightforward to show that $y(t) = \\lambda e^{st}$ fulfills above equation\n",
    "\n",
    "\\begin{equation}\n",
    "\\lambda e^{s t} e^{-s \\tau} = e^{-s \\tau} \\lambda e^{st}\n",
    "\\end{equation}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "An LTI system whose input/output relation is given by the following inhomogeneous linear ordinary differential equation (ODE) with constant coefficients is investigated\n",
    "\n",
    "\\begin{equation}\n",
    "a_0 y(t) + a_1 \\frac{d y(t)}{dt} + a_2 \\frac{d^2 y(t)}{dt^2} = x(t)\n",
    "\\end{equation}\n",
    "\n",
    "with $a_i \\in \\mathbb{R} \\quad \\forall i$. In the remainder, the output signal $y(t)$ of the system is computed by explicit solution of the ODE for $x(t) = e^{s t}$ as input signal. Integration constants are discarded for ease of illustration."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$\\displaystyle y{\\left(t \\right)} = \\frac{e^{s t}}{a_{0} + a_{1} s + a_{2} s^{2}}$"
      ],
      "text/plain": [
       "               s⋅t      \n",
       "              ℯ         \n",
       "y(t) = ─────────────────\n",
       "                       2\n",
       "       a₀ + a₁⋅s + a₂⋅s "
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "sym.init_printing()\n",
    "\n",
    "t, s, a0, a1, a2 = sym.symbols('t s a:3')\n",
    "x = sym.exp(s * t)\n",
    "y = sym.Function('y')(t)\n",
    "\n",
    "ode = sym.Eq(a0*y + a1*y.diff(t) + a2*y.diff(t, 2), x)\n",
    "solution = sym.dsolve(ode)\n",
    "solution.subs({'C1': 0, 'C2': 0})"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercises**\n",
    "\n",
    "* Is the complex exponential signal an eigenfunction of the system?\n",
    "* Introduce $x(t) = e^{s t}$ and $y(t) = \\lambda \\cdot e^{s t}$ into the ODE and solve manually for the eigenvalue $\\lambda$. How is the result related to above result derived by solving the ODE?\n",
    "* Can you generalize your findings to an ODE of arbitrary order?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "The following inhomogeneous linear ODE with time-dependent coefficient is considered as an example for a **time-variant** but linear system\n",
    "\n",
    "\\begin{equation}\n",
    "t \\cdot \\frac{d y(t)}{dt} = x(t)\n",
    "\\end{equation}\n",
    "\n",
    "The output signal $y(t)$ of the system for a complex exponential signal at the input $x(t) = e^{st}$ is computed by explicit solution of the ODE. Again integration constants are discarded."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$\\displaystyle y{\\left(t \\right)} = \\operatorname{Ei}{\\left(s t \\right)}$"
      ],
      "text/plain": [
       "y(t) = Ei(s⋅t)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ode = sym.Eq(t*y.diff(t), x)\n",
    "solution = sym.dsolve(ode)\n",
    "solution.subs('C1', 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note, $\\text{Ei}(\\cdot)$ denotes the [exponential integral](http://docs.sympy.org/latest/modules/functions/special.html#sympy.functions.special.error_functions.Ei). The response $y(t)$ of the time-variant system is not equal to a weighted complex exponential signal $\\lambda \\cdot e^{s t}$. It can be concluded that complex exponentials are no eigenfunctions of this particular time-variant system."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example**\n",
    "\n",
    "A final example considers the following non-linear inhomogeneous ODE with constant coefficients\n",
    "\n",
    "\\begin{equation}\n",
    "\\left( \\frac{d y(t)}{dt} \\right)^2 = x(t)\n",
    "\\end{equation}\n",
    "\n",
    "as example for a **non-linear** but time-invariant system. Again, the output signal $y(t)$ of the system for a complex exponential signal at the input $x(t) = e^{st}$ is computed by explicit solution of the ODE. As before, integration constants are discarded."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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XPXwih74JX/6swcA4MA4q9YzGaoAwaqXlprKV3vNJE0AdwoSh6b8DTlVRhJ3pvwqYnn6tjEPP/ArR1UkwAZ0mGsXtFK9QwH5f+OqEheNqvxD0b7nGWFxs6xKvhmlVW1BO5F9N4461rk2b8VdbPZ/c9Kbpf8Co1vgx/Q/ArSrJWZWn/RaPAMc/fImxm0XjRD3C+fpmjPt/M13ZtyGm/76I3cQ3/Q/D7SiVGYcjSFbhwY5jkMh3Fa0boZJaZNFj2MXXOiPA5yyXhYDpv2d/mP57AtYS/X5LuIMXgoAGPpMFSnucbyJy4+IolE/5K+xdH/Oo/WACU4COAgYl3p/lH5R4ZbPTuCRAhvtHwA3zRu4jBkx+4Bt3J9yCWolxFg99Ccr9RA8XIwXFStn0F3Uir5d6ELOTF/4oDmOTlnDCcKNfQXqYHPzQ+eFIgvpSHyRMxKNd+JGWcPILRm7SkjdxwQVzKf+kCyE36SrrHWL7Z3EIqM/oU9N/Rc8IG9N/houwmIT+B10AVdF39hqIgDo6Lgqd/mZb8VlAWBTvDSxyU8kyQgHDgqKo/LkOnYvEElMlNws8C21Y3MtAZGnwBt/wSbBsCBFF1M/lZjEPJvN/qxeUKoO/bBR8y4s0F6b8pIf6oIRJHWAIXulhwiOcLyNCPXN+tCd9vSQ36QvKw/KjXhxZBcZBNvlR98KXTHr/U0+4YEw27Wmtt+LYTICA+sP0fwecs/Fs+heGwmJ0+lcZ3S6AukOfOuk4CLAoVP2/xzilLT9XJo1fNKDTop5VmR01GtQslH3MA6VJ94jIHbFmx5c3lMunm6lcuVn8y1eIo8CKlCPWEYXGyJRcKwyJx6WeYBSGH0+SItyEhH+nTWVlfuSdNw/0Qv2ZQPKGz5yj6VrvGN/2shAw/Rf7w/R/i8dk9H92W6ZdK0GABSztRFdS51GqqQUSpgCx+7tyAQqLC35alMtxat7f1/inxThXLjv+sqHchzlPmIBYl4PS8p43XK4Gs0E7CEeiBGPB7azByI8+R+LQaBSPcQEzwR9RIW3hU2guVIvSjIhXl3o3luXA2RAw/WfQa1zH8Wz6FybCYzL6vz/b8HfBvRHQwGDxYoFpXUR6Z77OBGGxVdXLi3G+NX0lDvm0de5YLscUaYHPIrO7LzMfLOaVRumRRFB/jk+QknCdM0cXT/V8rQeJBMcWhHUx6DI81wMDgo4FUheORci/b72VxGYpCKgPTf/Fzojj2fR/i8sk9G/G4RbwNbjYbRw0gSSJA5OJnibCWUO7htYxiu6ZUOtMjHMULtyOLoM5ilTtEfNkV8/C3mba+gcdCJiEl7I/ZJnhx2LfJf+QROmZSK9kw2QERkNuGBtue4Wh6VtvJbFZEAKm/2JnxPFs+hcuovHJ6P+s2A9+WzgChfNNDRR2lZHrXnjVT189tZ8jABblgjIgJSksTLJycgYaTdsCHuM12rlyL6si5squCq7yY1GnH1/riYwCfhfKC4VL3F0M0pWCBETpOfYgT3QfIl6nqneXOjnO6RAw/eewzI1n0/8NLpPRvxmH3EBcgfNcdQxctogGLpvzaxaDPZuv1PhvhEP5SAIx/Q/yT9IZveNGgajOgG9XQ7ks7JFBCen0TrlxJ4Qf/cRTa5SGPiQN/Rn7N/rxlUY+v3I+5byfK37Zj/eIQ9d6l8vx+/wImP6P+8D0X8RkEvr355hF0Cd/0yTPjhjJQevnmIrLrpTdZ9hF6z1p/8tvtybDBdH8dQZCwEn+cbFM2MgvSiBYjF/qnf9Ugel4rod+IA/ScWxAPhwXwBwEf8VNXzvITXg4EpDNZ5bc5cCxAJ9Ikoa0+Twr741QnIPiI1kgXZQ44BekB7IL/az3WN/ItFDfJ3pgith9xeMOOUOdOFJJWMhdW28S2EyHgPrC9H9HuHPj2fQ/Mv0L6/A5phmHOw7auybvM3HctSynNwJGYFkImP6X1R+uTTMCkXE4a47mUCNgBIyAETACRsAI3CJgxuEWi7lcV1nBfc7X56qryzUCRuC0CJj+T4unc5sAATMOE4DcUkS8jOdIM7glnYONgBFYPwKm//X34e5aYMZh5i7XmVFUfEOJzsYIGIEdIWD631Fnb6ipZhyW0ZnxSmA0622MgBHYFwKm/3319+pba8ZhAV2oXQef9PHd/pA/ZVpAC1wFI2AEhiJg+h+KnNPNhYAZh7mQPy6Xi0y4W+B3TSTx+/zjWPYxAkZgiwiY/rfYqxttU7rHoaJ9XEbzeYW/vUZEQJjHC2GeyR31H0Ys0VkbASOwFARM/0vpCddDYxHFXS6bK5vHSBze60Gjv/ykG/LKqfw+HgLqLHDnH86arhkerwLO2QgYgdkQMP3PBr0LPkbgibzKfAHvv/0fu4rg6kuOkmEAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$\\displaystyle \\left[ y{\\left(t \\right)} = - \\begin{cases} \\frac{2 \\sqrt{e^{s t}}}{s} & \\text{for}\\: s \\neq 0 \\\\t & \\text{otherwise} \\end{cases}, \\  y{\\left(t \\right)} = \\begin{cases} \\frac{2 \\sqrt{e^{s t}}}{s} & \\text{for}\\: s \\neq 0 \\\\t & \\text{otherwise} \\end{cases}\\right]$"
      ],
      "text/plain": [
       "⎡        ⎛⎧     ______           ⎞         ⎧     ______           ⎤\n",
       "⎢        ⎜⎪    ╱  s⋅t            ⎟         ⎪    ╱  s⋅t            ⎥\n",
       "⎢        ⎜⎪2⋅╲╱  ℯ               ⎟         ⎪2⋅╲╱  ℯ               ⎥\n",
       "⎢y(t) = -⎜⎨───────────  for s ≠ 0⎟, y(t) = ⎨───────────  for s ≠ 0⎥\n",
       "⎢        ⎜⎪     s                ⎟         ⎪     s                ⎥\n",
       "⎢        ⎜⎪                      ⎟         ⎪                      ⎥\n",
       "⎣        ⎝⎩     t       otherwise⎠         ⎩     t       otherwise⎦"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ode = sym.Eq(y.diff(t)**2, x)\n",
    "solution = sym.dsolve(ode)\n",
    "[si.subs('C1', 0) for si in solution]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Obviously for this non-linear system complex exponential signals are no eigenfunctions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Transfer Function\n",
    "\n",
    "The complex eigenvalue $\\lambda$ constitutes the weight of a complex exponential signal $e^{st}$ (using complex frequency $s$) experiences when passing through an LTI system. It is commonly termed as [*transfer function*](https://en.wikipedia.org/wiki/Transfer_function) and is denoted by $H(s)=\\lambda(s)$. Using this definition, the output signal $y(t)$ of an LTI system for a complex exponential signal at the input reads\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) = \\mathcal{H} \\{ e^{st} \\} = H(s) \\cdot e^{st}\n",
    "\\end{equation}\n",
    "\n",
    "Note that the concept of the transfer function is directly linked to the linearity and time-invariance of a system. Only in this case, complex exponential signals are eigenfunctions of the system and $H(s)$ describes the properties of an LTI system with respect to these.\n",
    "\n",
    "Above equation can be rewritten in terms of the magnitude $| H(s) |$ and phase $\\varphi(s) = \\arg \\{ H(s) \\}$ of the complex transfer function $H(s)$\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) = | H(s) | \\cdot e^{s t + j \\varphi(s)}\n",
    "\\end{equation}\n",
    "\n",
    "The magnitude $| H(s) |$ provides the frequency dependent attenuation/amplification of the eigenfunction $e^{st}$ by the system, while $\\varphi(s)$ provides the introduced phase-shift."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Link between Transfer Function and Impulse Response\n",
    "\n",
    "In order to establish a link between the transfer function $H(s)$ and the impulse response $h(t)$, the output signal $y(t) = \\mathcal{H} \\{ x(t) \\}$ of an LTI system with input signal $x(t)$ is considered. It is given by convolving the input signal with the impulse response\n",
    "\n",
    "\\begin{equation}\n",
    "y(t) = x(t) * h(t) = \\int_{-\\infty}^{\\infty} x(t-\\tau) \\cdot h(\\tau) \\; d\\tau\n",
    "\\end{equation}\n",
    "\n",
    "For a complex exponential signal as input $x(t) = e^{st}$, the output of an LTI system is given as $y(t) = \\mathcal{H} \\{ e^{st} \\} = H(s) \\cdot e^{st}$. Introducing both signals into the convolution integral yields\n",
    "\n",
    "\\begin{equation}\n",
    "H(s) \\cdot e^{st} = \\int_{-\\infty}^{\\infty} e^{st} e^{-s \\tau} \\cdot h(\\tau) \\; d\\tau\n",
    "\\end{equation}\n",
    "\n",
    "which after canceling $e^{s t}$ (the integral depends not on $t$) results in\n",
    "\n",
    "\\begin{equation}\n",
    "H(s) = \\int_{-\\infty}^{\\infty} h(\\tau) \\cdot e^{-s \\tau}  \\; d\\tau\n",
    "\\end{equation}\n",
    "\n",
    "under the assumption that the integral converges.\n",
    "The transfer function $H(s)$ can be computed from the impulse response $h(t)$ by integrating over the impulse response multiplied with the complex exponential function $e^{- s \\tau}$. This constitutes an integral transformation, which is later introduced in more detail as [Laplace transform](https://en.wikipedia.org/wiki/Laplace_transform).\n",
    "Usually the temporal variable $t$ is then used\n",
    "\n",
    "\\begin{equation}\n",
    "H(s) = \\int_{-\\infty}^{\\infty} h(t) \\cdot e^{-s t}  \\; d t\n",
    "\\end{equation}\n",
    "rather than $\\tau$ which remained from the convolution integral calculus.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "nbsphinx": "hidden"
   },
   "source": [
    "**Copyright**\n",
    "\n",
    "This notebook is provided as [Open Educational Resource](https://en.wikipedia.org/wiki/Open_educational_resources). Feel free to use the notebook for your own purposes. The text is licensed under [Creative Commons Attribution 4.0](https://creativecommons.org/licenses/by/4.0/), the code of the IPython examples under the [MIT license](https://opensource.org/licenses/MIT). Please attribute the work as follows: *Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples*."
   ]
  }
 ],
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