SS_Inverse_Laplacet_Transform_Partial_Fraction.ipynb

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   "cell_type": "markdown",
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   "source": [
    "# Inver Laplace Transform by using SymPy\n",
    "\n",
    "## Repeasted Linear Factor 경우,\n",
    "\n",
    "sympy를 이용할 경우, 간단히 구할 수 있음.\n",
    "\n",
    "아래의 소스는 아래의 $X(s)$를 Partial fraction decomposition하는 것과 역변환하는 경우를 보여줌.\n",
    "\n",
    "$$\n",
    "X(s) = \\frac{2s^2-25s-33}{s^3-3s^2-9s-5}\n",
    "$$\n",
    "\n",
    "* SymPy에서 $\\theta(t)$는 Unit Step Function이며, 다른 이름으로는 Heaviside step function이라고도 불림.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 230,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\frac{5}{s + 1} + \\frac{1}{\\left(s + 1\\right)^{2}} - \\frac{3}{s - 5}$"
      ],
      "text/plain": [
       "5/(s + 1) + (s + 1)**(-2) - 3/(s - 5)"
      ]
     },
     "execution_count": 230,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "\n",
    "s,t = sym.symbols('s t')\n",
    "\n",
    "\n",
    "num = 2*(s**2) - 25*s -33\n",
    "dim = s**3 -3*(s**2)-9*s-5\n",
    "\n",
    "X = num/dim\n",
    "\n",
    "sym.apart(X)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 231,
   "metadata": {},
   "outputs": [],
   "source": [
    "# 꽤 시간이 걸림.\n",
    "X = sym.nsimplify(X)\n",
    "x = sym.inverse_laplace_transform(X,s,t)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 232,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(t - 3*exp(6*t) + 5)*exp(-t)*Heaviside(t)\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(t - 3 e^{6 t} + 5\\right) e^{- t} \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "(t - 3*exp(6*t) + 5)*exp(-t)*Heaviside(t)"
      ]
     },
     "execution_count": 232,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "print(x)\n",
    "x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Complex Conjugate Pole의 경우,\n",
    "\n",
    "$$\n",
    "X(s) = \\frac{4(s+1)^2}{s(s^2+2s+2)}\n",
    "$$\n",
    "\n",
    "* SymPy에서 $\\theta(t)$는 Unit Step Function이며, 다른 이름으로는 Heaviside step function이라고도 불림."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 233,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(1 - I)*(1 + I)*(exp(t) + sin(t) + cos(t))*exp(-t)*Heaviside(t)\n"
     ]
    },
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(1 - i\\right) \\left(1 + i\\right) \\left(e^{t} + \\sin{\\left(t \\right)} + \\cos{\\left(t \\right)}\\right) e^{- t} \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "(1 - I)*(1 + I)*(exp(t) + sin(t) + cos(t))*exp(-t)*Heaviside(t)"
      ]
     },
     "execution_count": 233,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "num = 4*((s+1)**2)\n",
    "dim = s*(s**2+2*s+2)\n",
    "\n",
    "X = num/dim\n",
    "x = sym.inverse_laplace_transform(X,s,t)\n",
    "print(x)\n",
    "x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 234,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\left(1 - i\\right) \\left(1 + i\\right) \\left(e^{t} + \\sqrt{2} \\sin{\\left(t + \\frac{\\pi}{4} \\right)}\\right) e^{- t} \\theta\\left(t\\right)$"
      ],
      "text/plain": [
       "(1 - I)*(1 + I)*(exp(t) + sqrt(2)*sin(t + pi/4))*exp(-t)*Heaviside(t)"
      ]
     },
     "execution_count": 234,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.trigsimp(sym.trigsimp(x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "참고로, $\\sin(t+\\pi/4) = \\cos(t-\\pi/4)$임."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "sym.sin(t+sym.pi/4).rewrite(sym.cos)"
   ]
  }
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