ZTF_Devlopment.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from sympy import *\n",
    "init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "n=symbols('n', integer=True)\n",
    "y=Function('y', integer=True)\n",
    "x=Function('x', integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAACIAAAAQBAMAAACB27BXAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARN3vMmYQu3aZIqtUic3QRwAsAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAdElEQVQYGWNgQAKsUz06kbhAJqtwAPMCFKFlTQz8CVAR8XIgSGDYwSCMooSBwYRhLguKENcHBsUjKCKMGxh6AxgYGJUFXFHEGdIjZhagiiQwLEcVYGBgKUMXkUYX4NmALnKSgQFVjEX1ap8AiipOhuCHKAIAfLAUWQGj+RoAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$z^{- n}$$"
      ],
      "text/plain": [
       " -n\n",
       "z  "
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "z=symbols('z')\n",
    "zkernal=z**(-n); zkernal"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "def ZTransDef(f, n, LB=-oo, UB=oo):\n",
    "    \"\"\"\n",
    "    Function to incapsolte the function f in the Z Transform\n",
    "    Args:\n",
    "        f: sympy expsersion to be ZTransformed\n",
    "        n: dummy index varible\n",
    "        LB: lower bound of the Z Transrom summation\n",
    "        UB: upper bound of the Z Transrom summation\n",
    "\n",
    "    \n",
    "    \"\"\"\n",
    "    return Sum(f*zkernal.subs(n, n), (n, LB, UB))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Raw testing"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- 3 y{\\left (n \\right )} + y{\\left (n + 1 \\right )} = 4$$"
      ],
      "text/plain": [
       "-3⋅y(n) + y(n + 1) = 4"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TestEq=Eq(y(n+1)-3*y(n), 4); TestEq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ y{\\left (0 \\right )} : 1\\right \\}$$"
      ],
      "text/plain": [
       "{y(0): 1}"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "BC={y(0):1}; BC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$3 \\cdot 3^{n} - 2$$"
      ],
      "text/plain": [
       "   n    \n",
       "3⋅3  - 2"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "KnownSol=rsolve(TestEq, y(n), BC); KnownSol"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\begin{cases} \\begin{cases} 0 & \\text{for}\\: 3 \\left|{\\frac{1}{z}}\\right| < 1 \\wedge \\frac{\\left|{z}\\right|}{3} < 1 \\\\3 \\sum_{n=-\\infty}^{\\infty} 3^{n} z^{- n} & \\text{otherwise} \\end{cases} & \\text{for}\\: \\left|{\\frac{1}{z}}\\right| < 1 \\wedge \\left|{z}\\right| < 1 \\\\\\begin{cases} - 2 \\sum_{n=-\\infty}^{\\infty} z^{- n} & \\text{for}\\: 3 \\left|{\\frac{1}{z}}\\right| < 1 \\wedge \\frac{\\left|{z}\\right|}{3} < 1 \\\\- 2 \\sum_{n=-\\infty}^{\\infty} z^{- n} + 3 \\sum_{n=-\\infty}^{\\infty} 3^{n} z^{- n} & \\text{otherwise} \\end{cases} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "⎧        ⎧                       │1│       │z│                                \n",
       "⎪        ⎪       0         for 3⋅│─│ < 1 ∧ ─── < 1                            \n",
       "⎪        ⎪                       │z│        3                                 \n",
       "⎪        ⎪                                                                    \n",
       "⎪        ⎪    ∞                                                               \n",
       "⎪        ⎪   ___                                                 │1│          \n",
       "⎪        ⎨   ╲                                               for │─│ < 1 ∧ │z│\n",
       "⎪        ⎪    ╲     n  -n                                        │z│          \n",
       "⎪        ⎪3⋅  ╱    3 ⋅z           otherwise                                   \n",
       "⎪        ⎪   ╱                                                                \n",
       "⎪        ⎪   ‾‾‾                                                              \n",
       "⎪        ⎪  n = -∞                                                            \n",
       "⎪        ⎩                                                                    \n",
       "⎪                                                                             \n",
       "⎪⎧              ∞                                                             \n",
       "⎨⎪             ___                                                            \n",
       "⎪⎪             ╲                                                              \n",
       "⎪⎪              ╲     -n                  │1│       │z│                       \n",
       "⎪⎪         -2⋅  ╱    z              for 3⋅│─│ < 1 ∧ ─── < 1                   \n",
       "⎪⎪             ╱                          │z│        3                        \n",
       "⎪⎪             ‾‾‾                                                            \n",
       "⎪⎪            n = -∞                                                          \n",
       "⎪⎨                                                                 otherwise  \n",
       "⎪⎪      ∞              ∞                                                      \n",
       "⎪⎪     ___            ___                                                     \n",
       "⎪⎪     ╲              ╲                                                       \n",
       "⎪⎪      ╲     -n       ╲     n  -n                                            \n",
       "⎪⎪- 2⋅  ╱    z   + 3⋅  ╱    3 ⋅z           otherwise                          \n",
       "⎪⎪     ╱              ╱                                                       \n",
       "⎪⎪     ‾‾‾            ‾‾‾                                                     \n",
       "⎩⎩    n = -∞         n = -∞                                                   \n",
       "\n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       " < 1\n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    \n",
       "    "
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(ZTransDef(KnownSol, n).doit(conds='none'))\n",
    "#the conds='none' needs to have a way for findging the min, max bounds\n",
    "#of the ROC"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$4 \\left(\\begin{cases} \\frac{z}{- z + 1} + \\frac{1}{1 - \\frac{1}{z}} & \\text{for}\\: \\left|{\\frac{1}{z}}\\right| < 1 \\wedge \\left|{z}\\right| < 1 \\\\\\sum_{n=-\\infty}^{\\infty} z^{- n} & \\text{otherwise} \\end{cases}\\right)$$"
      ],
      "text/plain": [
       "  ⎛⎧  z        1        │1│              ⎞\n",
       "  ⎜⎪────── + ─────  for │─│ < 1 ∧ │z│ < 1⎟\n",
       "  ⎜⎪-z + 1       1      │z│              ⎟\n",
       "  ⎜⎪         1 - ─                       ⎟\n",
       "  ⎜⎪             z                       ⎟\n",
       "  ⎜⎪                                     ⎟\n",
       "  ⎜⎪    ∞                                ⎟\n",
       "4⋅⎜⎨   ___                               ⎟\n",
       "  ⎜⎪   ╲                                 ⎟\n",
       "  ⎜⎪    ╲     -n                         ⎟\n",
       "  ⎜⎪    ╱    z            otherwise      ⎟\n",
       "  ⎜⎪   ╱                                 ⎟\n",
       "  ⎜⎪   ‾‾‾                               ⎟\n",
       "  ⎜⎪  n = -∞                             ⎟\n",
       "  ⎝⎩                                     ⎠"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTR=ZTransDef(TestEq.rhs, n).doit(conds='none'); ZTR\n",
    "#sympy needs to add conds='none' to sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} z^{- n} \\left(- 3 y{\\left (n \\right )} + y{\\left (n + 1 \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ∞                            \n",
       " ___                           \n",
       " ╲                             \n",
       "  ╲     -n                     \n",
       "  ╱    z  ⋅(-3⋅y(n) + y(n + 1))\n",
       " ╱                             \n",
       " ‾‾‾                           \n",
       "n = -∞                         "
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL=ZTransDef(TestEq.lhs, n); ZTL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} - 3 z^{- n} y{\\left (n \\right )} + \\sum_{n=-\\infty}^{\\infty} z^{- n} y{\\left (n + 1 \\right )}$$"
      ],
      "text/plain": [
       "  ∞                    ∞                \n",
       " ___                  ___               \n",
       " ╲                    ╲                 \n",
       "  ╲        -n          ╲     -n         \n",
       "  ╱    -3⋅z  ⋅y(n) +   ╱    z  ⋅y(n + 1)\n",
       " ╱                    ╱                 \n",
       " ‾‾‾                  ‾‾‾               \n",
       "n = -∞               n = -∞             "
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL=expand(ZTL); ZTL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ y{\\left (n \\right )}, \\quad y{\\left (n + 1 \\right )}\\right ]$$"
      ],
      "text/plain": [
       "[y(n), y(n + 1)]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTLFs=list(ZTL.atoms(Function)); ZTLFs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ \\left ( n\\right ), \\quad \\left ( n + 1\\right )\\right ]$$"
      ],
      "text/plain": [
       "[(n,), (n + 1,)]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTLFsArgs=[i.args for i in ZTLFs]; ZTLFsArgs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAXEAAAAVBAMAAABCu8NvAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAZpkQzYndq1TvMrsiRHaZuvfyAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEuElEQVRIDdWXTYgcRRTH/z09Xzsf6yAGDEgykQRBIQ4xGGRB5zBREMVRFsWDywghJBBwT4FADoMHJSBmwD0JwfHiJYfd4CGHFexc/EDQuRjx4jaL4C0ZdEMwuup7r17VVHXP5u6DqXr1/tW//s90dXUPgOgQ/mdRO9phx/HJme8bmn4wK/mZytXELwa5BQRFGeyBFK3Rku4+XNIVLZzmx3xANJXDuIk3NK0kmnDxwr8XgUt/pjOZxvPDAfKyjxR1xsWSTt+TSzqjV/aNYTiH+YjSgFuJYqJJ3NWEu9oONcUhfRKYOK59rnMzQqX6EZ25G9Y8Ll5SaU8u6YQuDuNzynmdj/CcP8hjiRWbcP9OArxFvZNLKY3mhZshYjU1c0688QclAVIEy43aMiQrqSYfau91hN4PLCsn6/ycm/mMyyi53Uc0pN7J8ciXvdzNkJp1joidB0jRLbecyJCui+XOcU7o68DljuFknDcGSqBrM3Ep8NwIv9DQk1/2VC/1ZnA1dB4g5SDl4iuHsNy8c0bfBbaGxppzHm+n+Laz2AZ+vPg95ShRCpRXuUV9t9anjmXcvFJrAa9RFkQOwGro3Ec2z04+dVxs01xTsVzrnKsOHf9OzlNj7U3mV7pktDnCbdTJU38zjf+mDYeKdIGlRfXuMR6xHF/bfoLSh3jsRw7AYujcRy4deHSDdMPFNZprKpZrnXPVoRt3gPXUWDvI/Etj4JtCF5soD1EbnkLjHkEHpLj4ZJVTkhGjQafEezz2IwdgMXTuIzv4jicYLs5QaiqWa51z1aEb9Juv9421Yhu4yndVZyHFP1iYII53ZYOvTanq4ssOpyRTXOV8nTM/cgCUer3n13q9VZ4ld2iArH3OdcPFFc6lItzlXu9sr/cCF7k6Q5vVYjjnU+DnMc3Y6tTuibXqVJZPcBqs0Qx13hQrOec5AB8Q/uYB8ifWlWucS8Vy7W8OrjpvdIdepmUxpVqRbVQ2qDmMxZEsh0IXD7Q6aA6oaIMXGAWvFrzPGW5K6zdZAGuhcx9Zle+vXFktpmK51rlUHXoTOKjWZJ3Lk+g66qtyCy70ac+k/bvr2Yp2ZVBvkZk2KjSwd9JsUhbASujcR74LkHflyh1qKpZrnUvVoc2TSDiyt4jzx7DVkv3mhwmWozEKfT617i31KQ9E/mzfAd5zjyDekZprsgAWnPNFfhJ5yNrbr/46oc3KcHmP1coR5alzU3Xo8jCmNy3huP0cJ249mcgSoev19cO0jsbMMPt55ZW/Uh7x5W5Ha3RKvAic7nDNRRbAgnVeuLBzKEAWsJ++v+Xy2jMV5kqoc1N16Hjlt7FyZs55pXuPdzxtCGHrnu3xgGyFzrMAPtI65xzzkaKUE+lozx1oYleLDgNvYs04H/HTiHZD75VqzuuRL5dSWjtKNV0OwOU4+HL5Ny5LiNqaMVfiKe25y6KFI875/bzcqqc0qZ5Qw5F/JeWqk4/T4FmuuMgCnOCS+Ugjf6GzmJuLDNpw+CoAJ2kJ33qcM/fHwHnkqgsn85larsxJFhCIMpiPNPOWdLr9BsHRGbRwmvLwQpVuH40b2j9iC2GvcpSEZW9kAV5J0z2Qouq/uftwaZqimSP/Q/8DUaxg3+kpzYYAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\left \\{ y{\\left (n \\right )} : Y{\\left (z \\right )}, \\quad y{\\left (n + 1 \\right )} : z Y{\\left (z \\right )} + z y{\\left (0 \\right )}\\right \\}$$"
      ],
      "text/plain": [
       "{y(n): Y(z), y(n + 1): z⋅Y(z) + z⋅y(0)}"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Y=Function('Y')\n",
    "sups=[Y(z), z*Y(z)+z*y(0)]\n",
    "Sups={i:j for i, j in zip(ZTLFs, sups)}; Sups"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "scrolled": true
   },
   "outputs": [],
   "source": [
    "#ZTL.subs(y(n+1), z*Y(z)/zkernal)\n",
    "#sympy I have no Idea why this exspection was raised"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} \\left(- 3 y{\\left (n \\right )} + y{\\left (n + 1 \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ∞                        \n",
       " ___                       \n",
       " ╲                         \n",
       "  ╲    (-3⋅y(n) + y(n + 1))\n",
       "  ╱                        \n",
       " ╱                         \n",
       " ‾‾‾                       \n",
       "n = -∞                     "
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL=simplify(ZTL/zkernal); ZTL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} \\left(- 3 Y{\\left (z \\right )} + y{\\left (n + 1 \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ∞                        \n",
       " ___                       \n",
       " ╲                         \n",
       "  ╲    (-3⋅Y(z) + y(n + 1))\n",
       "  ╱                        \n",
       " ╱                         \n",
       " ‾‾‾                       \n",
       "n = -∞                     "
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL=ZTL.subs(y(n), Y(z)).doit(); ZTL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} \\left(z Y{\\left (z \\right )} - z y{\\left (0 \\right )} - 3 Y{\\left (z \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ∞                              \n",
       " ___                             \n",
       " ╲                               \n",
       "  ╲    (z⋅Y(z) - z⋅y(0) - 3⋅Y(z))\n",
       "  ╱                              \n",
       " ╱                               \n",
       " ‾‾‾                             \n",
       "n = -∞                           "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL=ZTL.subs(y(n+1), z*Y(z)-z*y(0)); ZTL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\infty z Y{\\left (z \\right )} - \\infty z y{\\left (0 \\right )} - \\infty Y{\\left (z \\right )}$$"
      ],
      "text/plain": [
       "∞⋅z⋅Y(z) - ∞⋅z⋅y(0) - ∞⋅Y(z)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABMAAAALBAMAAABv+6sJAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEKvvZom7mXYyzVQi3UQ6SGZXAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAbUlEQVQIHWNgYBBgAAIQwaj82YGBIayogYGBbQLHLwapDQwnGBg8GRj2J6xkYGA5wACUYP0LJBgcQEyGfAcg0QBmJv8DandgcGRg4IgxPsDAcYCBnYEhnGGyHYMrUKJcbQMDY71xAFA5BABtAwBKehUH3PvDFwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$\\infty$$"
      ],
      "text/plain": [
       "∞"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(1, (n, -oo, oo)).doit()\n",
    "#sympy: this should be 1 since there is no index inside the sum"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$z Y{\\left (z \\right )} - z y{\\left (0 \\right )} - 3 Y{\\left (z \\right )}$$"
      ],
      "text/plain": [
       "z⋅Y(z) - z⋅y(0) - 3⋅Y(z)"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL=simplify(ZTL).subs(Sum(1, (n, -oo,oo)), 1).doit(); ZTL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$z Y{\\left (z \\right )} - z y{\\left (0 \\right )} - 3 Y{\\left (z \\right )} = 4 \\left(\\begin{cases} \\frac{z}{- z + 1} + \\frac{1}{1 - \\frac{1}{z}} & \\text{for}\\: \\left|{\\frac{1}{z}}\\right| < 1 \\wedge \\left|{z}\\right| < 1 \\\\\\sum_{n=-\\infty}^{\\infty} z^{- n} & \\text{otherwise} \\end{cases}\\right)$$"
      ],
      "text/plain": [
       "                             ⎛⎧  z        1        │1│              ⎞\n",
       "                             ⎜⎪────── + ─────  for │─│ < 1 ∧ │z│ < 1⎟\n",
       "                             ⎜⎪-z + 1       1      │z│              ⎟\n",
       "                             ⎜⎪         1 - ─                       ⎟\n",
       "                             ⎜⎪             z                       ⎟\n",
       "                             ⎜⎪                                     ⎟\n",
       "                             ⎜⎪    ∞                                ⎟\n",
       "z⋅Y(z) - z⋅y(0) - 3⋅Y(z) = 4⋅⎜⎨   ___                               ⎟\n",
       "                             ⎜⎪   ╲                                 ⎟\n",
       "                             ⎜⎪    ╲     -n                         ⎟\n",
       "                             ⎜⎪    ╱    z            otherwise      ⎟\n",
       "                             ⎜⎪   ╱                                 ⎟\n",
       "                             ⎜⎪   ‾‾‾                               ⎟\n",
       "                             ⎜⎪  n = -∞                             ⎟\n",
       "                             ⎝⎩                                     ⎠"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZT=Eq(ZTL, ZTR); ZT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$Y{\\left (z \\right )} = \\begin{cases} \\frac{z y{\\left (0 \\right )}}{z - 3} & \\text{for}\\: \\left|{\\frac{1}{z}}\\right| < 1 \\wedge \\left|{z}\\right| < 1 \\\\\\frac{1}{z - 3} \\left(z y{\\left (0 \\right )} + 4 \\sum_{n=-\\infty}^{\\infty} z^{- n}\\right) & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "       ⎧       z⋅y(0)              │1│              \n",
       "       ⎪       ──────          for │─│ < 1 ∧ │z│ < 1\n",
       "       ⎪       z - 3               │z│              \n",
       "       ⎪                                            \n",
       "       ⎪             ∞                              \n",
       "       ⎪            ___                             \n",
       "       ⎪            ╲                               \n",
       "Y(z) = ⎨             ╲     -n                       \n",
       "       ⎪z⋅y(0) + 4⋅  ╱    z                         \n",
       "       ⎪            ╱                               \n",
       "       ⎪            ‾‾‾                             \n",
       "       ⎪           n = -∞                           \n",
       "       ⎪─────────────────────        otherwise      \n",
       "       ⎪        z - 3                               \n",
       "       ⎩                                            "
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTF=Eq(Y(z), solve(ZT, Y(z))[0]); ZTF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$Y{\\left (z \\right )} = \\begin{cases} \\frac{z}{z - 3} & \\text{for}\\: \\left|{\\frac{1}{z}}\\right| < 1 \\wedge \\left|{z}\\right| < 1 \\\\\\frac{z + 4 \\sum_{n=-\\infty}^{\\infty} z^{- n}}{z - 3} & \\text{otherwise} \\end{cases}$$"
      ],
      "text/plain": [
       "       ⎧       z              │1│              \n",
       "       ⎪     ─────        for │─│ < 1 ∧ │z│ < 1\n",
       "       ⎪     z - 3            │z│              \n",
       "       ⎪                                       \n",
       "       ⎪        ∞                              \n",
       "       ⎪       ___                             \n",
       "       ⎪       ╲                               \n",
       "Y(z) = ⎨        ╲     -n                       \n",
       "       ⎪z + 4⋅  ╱    z                         \n",
       "       ⎪       ╱                               \n",
       "       ⎪       ‾‾‾                             \n",
       "       ⎪      n = -∞                           \n",
       "       ⎪────────────────        otherwise      \n",
       "       ⎪     z - 3                             \n",
       "       ⎩                                       "
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTF=ZTF.subs(BC); ZTF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} z^{- n} \\left(- 3 y{\\left (n \\right )} + y{\\left (n + 1 \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ∞                            \n",
       " ___                           \n",
       " ╲                             \n",
       "  ╲     -n                     \n",
       "  ╱    z  ⋅(-3⋅y(n) + y(n + 1))\n",
       " ╱                             \n",
       " ‾‾‾                           \n",
       "n = -∞                         "
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL=ZTransDef(TestEq.lhs, n); ZTL"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Devlopement of Shifting"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [],
   "source": [
    "k=symbols('k', integer=True)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$y{\\left (0 \\right )}$$"
      ],
      "text/plain": [
       "y(0)"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(y(k)*zkernal.subs(n, k), (k, 0, 0)).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$y{\\left (0 \\right )} + \\frac{1}{z} y{\\left (1 \\right )}$$"
      ],
      "text/plain": [
       "       y(1)\n",
       "y(0) + ────\n",
       "        z  "
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(y(k)*zkernal.subs(n, k), (k, 0, 1)).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAAoAAAAOBAMAAADkjZCYAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAiXZmMs1UEN0i77urRJlR0qN3AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAAUUlEQVQIHWNgYFQWYWBgCGOomMDAvICBMYCB+wAD23cG/gMMvN8Y6h8w8H5imC/AwAIkHzCwfISKAGXZvjFwb2Bg/g7VxdDGUOXAwFCodIQBAG3HFgUteuAKAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$0$$"
      ],
      "text/plain": [
       "0"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(y(k)*zkernal.subs(n, k), (k, 0, -1)).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- z y{\\left (-1 \\right )}$$"
      ],
      "text/plain": [
       "-z⋅y(-1)"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sum(y(k)*zkernal.subs(n, k), (k, 0, -2)).doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$z \\left(Y{\\left (z \\right )} - y{\\left (0 \\right )}\\right)$$"
      ],
      "text/plain": [
       "z⋅(Y(z) - y(0))"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ShiftBound=1\n",
    "z**(ShiftBound)*(Y(z)-Sum(y(k)*zkernal.subs(n, k), (k, 0, ShiftBound-1)).doit())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAOMAAAAUBAMAAACNN5feAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMARN3vMmYQu3aZIqtUic3QRwAsAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADaklEQVRIDXVWS2gTURQ90yaTNE1s0I3gogHpRrqom+LKBkFcSQNuVPyEdlGhaLupii4aEKHZaBaKilgHRdwILSgiLkpxpy7sQkFxE8GVFFHUUqy/c97Mm7yO6YX73pl7zzt35s2dlwDWchWL4EDgXhjOB3F6HYi4Sm9EsRIvuFAe21iMMIbxobUA+aFdgD8ZxZ+28i7issLF14DSG1CsRHcAyGP7FCMQDjd5uUzfFHCQnQqn5EjuMkYCk96AIon8IO++GbqVSDcsgmDHT959kaFDNlyoWeTO5KYHUGhC6fYUSYy/X+Gqt5FHAh1BBFiOMPsbOKLIDhv2ByxyZ3Iz88ivQOn2FCORVsnRyCOBOy0hQf8bvBrnXIlDaB8scGdye+bh/QKUbksxEqbkJm6b3Nte3Et2ne5ATBf3Szrb4LB5Z6oCvNN1y05cR25QyyYa8FbDdILi12u4WzUSpmSBYvKxgzcmKTRFdyBePWYOyFT4xFP1a4THdN2ypS3IftGyuSWkWFLpBKXgDeCykYApmW4C8iruS6ef7kB0rimKjkWWRE439UzXseUWZ5GZ1LK5BlJ880qvp+BBVxNnjERYMl9i69KROs0BOzU4MKP3DXQWNe6rchim+5+NlXUfu9BT07JoY5WW45ShfCSqdtawGkqYp0x9YQk6tokXlnRgZ9OETUlvXtjomaAZvBUMl7WM7ZNn+8QlWxT0VlPf25TMGz2zsQ7UG5JpY/FSCJvNGA/ctAvmfXB79QxKJyiYRXbA2VivxBalPwdUVO3jQBwvMhC2T76BbsJEb/BzPG+WhUdBu/YhgSx14Lr2SfW9uSR19rcDgWnGaIUGsOfoQX1wt0wgHrrmvR/hp9GP0XKYTlBwE70VI4GsWqNryXgXDpgPmHviQCz8PWvEvRLQSO/WTakhHPOf1HkoaitPzjziqHSCgvGZ2wG8EqUf/rnKY6AcOkLTKReZA90Dr2TzdubbNIejufZL7GJ6wmZ5HZ+ZI7yQR5Y41m1442N9ARN89nhZofb/sd7d9L9SiMd6aIljHedsAi5EJojiyV+mPpitt8uUTlI6Kpkaw1bCb3Ij6LGNxYg/0S2zv6/r7kPprfWyJstVaVtecZk3c0WTlVBpW15x999Huz8i6UCk/y3ipgNuMr2tRf9lDjNJ/wdBQeaRRKMdQwAAAABJRU5ErkJggg==\n",
      "text/latex": [
       "$$z \\left(z \\left(Y{\\left (z \\right )} - y{\\left (0 \\right )}\\right) - y{\\left (1 \\right )}\\right)$$"
      ],
      "text/plain": [
       "z⋅(z⋅(Y(z) - y(0)) - y(1))"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def zShift(z, y, Y, ShiftBound):\n",
    "    \"\"\"\n",
    "    Functiont to generate the Right and Left Z Shift property of Z transfrom\n",
    "    Args:\n",
    "        z(sympy symbol): Z transfrom z symbol\n",
    "        y(sympy Function): discrate \"time\" Function being transformed from\n",
    "        Y(sympy Function): discrate Z domian Function that y transfroms into\n",
    "        \n",
    "    \"\"\"\n",
    "    return simplify(z**(ShiftBound)*(Y(z)-Sum(y(k)*zkernal.subs(n, k), (k, 0, ShiftBound-1)).doit()))\n",
    "\n",
    "zShift(z, y,Y, 2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAO0AAAA5BAMAAAAozLGFAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaZVCKJq0S772Yyzd3BNagQAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAEZElEQVRYCe1YTYgcRRT+eqZ7/np2plUkuJCk2QUPHuJqNAox7CCoF3HH4A8YSIYo5pRlEQKyIgyIQiK4iyi65LI5RHJQMqDe1AziwZsjIhF0kjkYkUBg16ywGGP76u9V1wwelrANwjx4U9/76lV/1a+ru3oaYLv9AeDgSxxmBbxXT54NH6zGWekZnRywOA28aeKs2hCYWAf8rPSMDukWrwHfmjir1u/i4V3fezNZ6bHO43u7mL8YcTwG4wqMK7C1CkwkKYu3NvZWsv3kbzX8zrcXk9VbOdIWx87+wwO+2mC4/aCWLLDIAUbbD6rJDRbJdxluP7iQNFmkwyiwkDkChvbtmHQ3cMgNU9FjhIUbKycDA1PtXRY/9/J7HDB9gikHeMtOmA6KNFXhxrzz1w1MtT8yDmYw1zER0zsN47YTqQOnevwzgNdTzvRS0mBsQDhjEHIbqA90ZOlCrKlHdauaT53Ij1W44xi9z+CodsUBhYRmM2Q1O+/SOdRpqtIs7a1qytW9olnVGF2EQndSO6dcvsnQgMMGyJbrnKJ/0hmObjBwxrm6E216kSNnm0xajDWg90tr3m/wpmO8EIHo0rFnp+6nvhd1v9ZVCTm6PG+dOEjJ0lzdAnUKZ/OTTcYanE4Rwfx9KJRWcS9AdKH8LnY3gS91htZVCeUO0FqJvT9Vp6sb9gDh1mb/imwg0a9OfE/7mXwPKwDR39R6ONIFntIZWlcl1BqoNq4h0M98V9cf0MsyubXaSKHP0ZrvS+tSWuVMVInxB0B0tLuNC8R9J4Yf7/ev9vsXCamEShuet4lwjZhCv3/pk35/QeTJdVUlVri1wO4NmiQBY0ET5Q0sRVU6CUHPAa9TI3Wp1eerEkgX/hqKPeogc893VDc8q/Lsb6rO9XWhewC5VVlnQrjhAe/oZKMrE6jOoCtS70Sy19UtDWhVkltb6lqsUGpdVZZRW8cHKC/IdYWP4K8XRtaVShDrqtLCXNSSh3F1R9ZVdXTfNXcJDQ874jn5EJbomEQHmyj0vgD2qQlynWWCuE2OtHE87Lq6OfHcyLeUE1Q2GRvEramiIN7Y+yGwY2q2KYtLlQpeawC/6FxTZ5kgykhDn/9MdZrzzS/e3E/PjK5yPRD4gRED+0Bkiq4rLO0NdIfRhbjwuKJZ1RhdFe2iRrix8rJBtg2pYGkr9uj2oJozzfvCbSpNJ7j7ghelj3GUAuHG5m1vw3B4n5EEtU45FoDpnZK2Pzqh3LTUEPJ67j6YuonCBufyBq+Y0tQjEjDNE9AjdIK3zEcYBmJK6WmlbqJTNte80FhGIkOH/3Veh4YG2PBzgsK1BfYmuuO6ITNoJ1tmAk+YV/gMVIHL6vF/6Tz9XxlkoihF8qm/R0mcne5YaVyBcQX+jxUI9389PTTvTD68h6c6lWZxhey0ls/mw/vTe1CPnBPOZfPh/RXwfntSnHaU0Yf3j7GvWhS70c/6rLP58B6s4fcnnTr7mXx4Ly3g7o6jiyw+vP8L0EQpfJs+jkAAAAAASUVORK5CYII=\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} \\left(- 3 y{\\left (n \\right )} + y{\\left (n + 1 \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ∞                        \n",
       " ___                       \n",
       " ╲                         \n",
       "  ╲    (-3⋅y(n) + y(n + 1))\n",
       "  ╱                        \n",
       " ╱                         \n",
       " ‾‾‾                       \n",
       "n = -∞                     "
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL=simplify(ZTL/zkernal); ZTL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} \\left(z \\left(Y{\\left (z \\right )} - y{\\left (0 \\right )}\\right) - 3 Y{\\left (z \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ∞                              \n",
       " ___                             \n",
       " ╲                               \n",
       "  ╲    (z⋅(Y(z) - y(0)) - 3⋅Y(z))\n",
       "  ╱                              \n",
       " ╱                               \n",
       " ‾‾‾                             \n",
       "n = -∞                           "
      ]
     },
     "execution_count": 34,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL=ZTL.subs(y(n), zShift(z, y,Y, 0)).subs(y(n+1), zShift(z, y,Y, 1))\n",
    "ZTL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$z Y{\\left (z \\right )} - z y{\\left (0 \\right )} - 3 Y{\\left (z \\right )}$$"
      ],
      "text/plain": [
       "z⋅Y(z) - z⋅y(0) - 3⋅Y(z)"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(ZTL).subs(Sum(1, (n, -oo, oo)), 1).doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# AutoGen Shift"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} z^{- n} \\left(- 3 y{\\left (n \\right )} + y{\\left (n + 1 \\right )}\\right)$$"
      ],
      "text/plain": [
       "  ∞                            \n",
       " ___                           \n",
       " ╲                             \n",
       "  ╲     -n                     \n",
       "  ╱    z  ⋅(-3⋅y(n) + y(n + 1))\n",
       " ╱                             \n",
       " ‾‾‾                           \n",
       "n = -∞                         "
      ]
     },
     "execution_count": 36,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL=ZTransDef(TestEq.lhs, n); ZTL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left [ y{\\left (n \\right )}, \\quad y{\\left (n + 1 \\right )}\\right ]$$"
      ],
      "text/plain": [
       "[y(n), y(n + 1)]"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTLFs=list(ZTL.atoms(Function)); ZTLFs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( y{\\left (n \\right )}, \\quad y{\\left (n + 1 \\right )}\\right )$$"
      ],
      "text/plain": [
       "(y(n), y(n + 1))"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F1=ZTLFs[0]; F2=ZTLFs[1]; F1, F2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "There should be a way get get $y$ from $y(n)$ but I cant figure it out; would then like to have a way to create $Y$ from $y$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( 0, \\quad 1\\right )$$"
      ],
      "text/plain": [
       "(0, 1)"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "F1.args[0].subs(n, 0), F2.args[0].subs(n, 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( Y{\\left (z \\right )}, \\quad z \\left(Y{\\left (z \\right )} - y{\\left (0 \\right )}\\right)\\right )$$"
      ],
      "text/plain": [
       "(Y(z), z⋅(Y(z) - y(0)))"
      ]
     },
     "execution_count": 40,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "zShift(z, y, Y, 0), zShift(z, y, Y, 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ y{\\left (n \\right )} : Y{\\left (z \\right )}, \\quad y{\\left (n + 1 \\right )} : z \\left(Y{\\left (z \\right )} - y{\\left (0 \\right )}\\right)\\right \\}$$"
      ],
      "text/plain": [
       "{y(n): Y(z), y(n + 1): z⋅(Y(z) - y(0))}"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Sups={i: zShift(z, y, Y, i.args[0].subs(n, 0)) for i in ZTLFs}\n",
    "Sups"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Working Z Transfrom Process Test"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Set up discrate \"time\" equation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$- n y{\\left (n - 3 \\right )} + 3 n y{\\left (n + 2 \\right )} - 7 n + y{\\left (n \\right )} = 8 n - x{\\left (n \\right )} + 2 x{\\left (n + 1 \\right )}$$"
      ],
      "text/plain": [
       "-n⋅y(n - 3) + 3⋅n⋅y(n + 2) - 7⋅n + y(n) = 8⋅n - x(n) + 2⋅x(n + 1)"
      ]
     },
     "execution_count": 42,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TestEq2=Eq(3*n*y(n+2)-n*y(n-3)+y(n)-7*n, 2*x(n+1)-x(n)+8*n); TestEq2"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "call `ZTransDef` on each side respecftly, assuming that each side contains a single function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\sum_{n=-\\infty}^{\\infty} z^{- n} \\left(- n y{\\left (n - 3 \\right )} + 3 n y{\\left (n + 2 \\right )} - 7 n + y{\\left (n \\right )}\\right), \\quad \\sum_{n=-\\infty}^{\\infty} z^{- n} \\left(8 n - x{\\left (n \\right )} + 2 x{\\left (n + 1 \\right )}\\right)\\right )$$"
      ],
      "text/plain": [
       "⎛  ∞                                                     ∞                    \n",
       "⎜ ___                                                   ___                   \n",
       "⎜ ╲                                                     ╲                     \n",
       "⎜  ╲     -n                                              ╲     -n             \n",
       "⎜  ╱    z  ⋅(-n⋅y(n - 3) + 3⋅n⋅y(n + 2) - 7⋅n + y(n)),   ╱    z  ⋅(8⋅n - x(n) \n",
       "⎜ ╱                                                     ╱                     \n",
       "⎜ ‾‾‾                                                   ‾‾‾                   \n",
       "⎝n = -∞                                                n = -∞                 \n",
       "\n",
       "             ⎞\n",
       "             ⎟\n",
       "             ⎟\n",
       "             ⎟\n",
       "+ 2⋅x(n + 1))⎟\n",
       "             ⎟\n",
       "             ⎟\n",
       "             ⎠"
      ]
     },
     "execution_count": 43,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTL=ZTransDef(TestEq2.lhs, n); ZTR=ZTransDef(TestEq2.rhs, n)\n",
    "ZTL, ZTR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left ( \\left [ y{\\left (n + 2 \\right )}, \\quad y{\\left (n - 3 \\right )}, \\quad y{\\left (n \\right )}\\right ], \\quad \\left [ x{\\left (n + 1 \\right )}, \\quad x{\\left (n \\right )}\\right ]\\right )$$"
      ],
      "text/plain": [
       "([y(n + 2), y(n - 3), y(n)], [x(n + 1), x(n)])"
      ]
     },
     "execution_count": 44,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ZTLFs=list(ZTL.atoms(Function)); ZTRFs=list(ZTR.atoms(Function)); \n",
    "ZTLFs, ZTRFs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ y{\\left (n \\right )} : Y{\\left (z \\right )}, \\quad y{\\left (n - 3 \\right )} : y{\\left (-3 \\right )} + \\frac{1}{z} y{\\left (-2 \\right )} + \\frac{1}{z^{2}} y{\\left (-1 \\right )} + \\frac{1}{z^{3}} Y{\\left (z \\right )}, \\quad y{\\left (n + 2 \\right )} : z \\left(z \\left(Y{\\left (z \\right )} - y{\\left (0 \\right )}\\right) - y{\\left (1 \\right )}\\right)\\right \\}$$"
      ],
      "text/plain": [
       "⎧                              y(-2)   y(-1)   Y(z)                           \n",
       "⎪y(n): Y(z), y(n - 3): y(-3) + ───── + ───── + ────, y(n + 2): z⋅(z⋅(Y(z) - y(\n",
       "⎨                                z        2      3                            \n",
       "⎪                                        z      z                             \n",
       "⎩                                                                             \n",
       "\n",
       "           ⎫\n",
       "0)) - y(1))⎪\n",
       "           ⎬\n",
       "           ⎪\n",
       "           ⎭"
      ]
     },
     "execution_count": 45,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "SupsL={i: zShift(z, y, Y, i.args[0].subs(n, 0)) for i in ZTLFs}\n",
    "SupsL"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ x{\\left (n \\right )} : X{\\left (z \\right )}, \\quad x{\\left (n + 1 \\right )} : z \\left(X{\\left (z \\right )} - x{\\left (0 \\right )}\\right)\\right \\}$$"
      ],
      "text/plain": [
       "{x(n): X(z), x(n + 1): z⋅(X(z) - x(0))}"
      ]
     },
     "execution_count": 46,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "X=Function('X')\n",
    "SupsR={i: zShift(z, x, X, i.args[0].subs(n, 0)) for i in ZTRFs}\n",
    "SupsR"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 47,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\left \\{ x{\\left (n \\right )} : z^{n} X{\\left (z \\right )}, \\quad x{\\left (n + 1 \\right )} : z z^{n} \\left(X{\\left (z \\right )} - x{\\left (0 \\right )}\\right)\\right \\}$$"
      ],
      "text/plain": [
       "⎧       n                    n              ⎫\n",
       "⎨x(n): z ⋅X(z), x(n + 1): z⋅z ⋅(X(z) - x(0))⎬\n",
       "⎩                                           ⎭"
      ]
     },
     "execution_count": 47,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "RSups={}\n",
    "for i in expand(ZTR).args:\n",
    "    if list(i.atoms(Function))!=[]:\n",
    "        key=list(i.atoms(Function))[0]\n",
    "        val=zShift(z, x, X, list(list(i.atoms(Function))[0].args)[0].subs(n, 0))\n",
    "\n",
    "        RSups[key]=val/zkernal\n",
    "RSups"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Linch Pin\n",
    "if this `ValueError: substitution can not create dummy dependencies\n",
    "` in starting on line 320 in `sympy/concrete/expr_with_limits.py` this would be done"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 48,
   "metadata": {},
   "outputs": [],
   "source": [
    "#ZTR.subs(RSups)\n",
    "#!!!!!!!!if this value Error was not being trigured this would be done"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# ZTransfrom Test on common functions listed no Z Tranfrom tables\n",
    "\n",
    "https://en.wikipedia.org/wiki/Z-transform#Table_of_common_Z-transform_pairs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Delta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 49,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} z^{- n} \\delta\\left(n\\right)$$"
      ],
      "text/plain": [
       "  ∞            \n",
       " ___           \n",
       " ╲             \n",
       "  ╲     -n     \n",
       "  ╱    z  ⋅δ(n)\n",
       " ╱             \n",
       " ‾‾‾           \n",
       "n = -∞         "
      ]
     },
     "execution_count": 49,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TF=ZTransDef(DiracDelta(n), n); TF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 50,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} z^{- n} \\delta\\left(n\\right)$$"
      ],
      "text/plain": [
       "  ∞            \n",
       " ___           \n",
       " ╲             \n",
       "  ╲     -n     \n",
       "  ╱    z  ⋅δ(n)\n",
       " ╱             \n",
       " ‾‾‾           \n",
       "n = -∞         "
      ]
     },
     "execution_count": 50,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TF.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Step Function"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 58,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} z^{- n} \\theta\\left(n\\right)$$"
      ],
      "text/plain": [
       "  ∞                    \n",
       " ___                   \n",
       " ╲                     \n",
       "  ╲     -n             \n",
       "  ╱    z  ⋅Heaviside(n)\n",
       " ╱                     \n",
       " ‾‾‾                   \n",
       "n = -∞                 "
      ]
     },
     "execution_count": 58,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TF=ZTransDef(Heaviside(n), n); TF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} z^{- n} \\theta\\left(n\\right)$$"
      ],
      "text/plain": [
       "  ∞                    \n",
       " ___                   \n",
       " ╲                     \n",
       "  ╲     -n             \n",
       "  ╱    z  ⋅Heaviside(n)\n",
       " ╱                     \n",
       " ‾‾‾                   \n",
       "n = -∞                 "
      ]
     },
     "execution_count": 59,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TF.doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Piecewise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 61,
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} \\begin{cases} 0 & \\text{for}\\: n < 0 \\\\z^{- n} \\theta\\left(0\\right) & \\text{for}\\: n = 0 \\\\z^{- n} & \\text{for}\\: n > 0 \\end{cases}$$"
      ],
      "text/plain": [
       "  ∞                                \n",
       "______                             \n",
       "╲                                  \n",
       " ╲     ⎧       0          for n < 0\n",
       "  ╲    ⎪                           \n",
       "   ╲   ⎪ -n                        \n",
       "    ╲  ⎨z  ⋅Heaviside(0)  for n = 0\n",
       "    ╱  ⎪                           \n",
       "   ╱   ⎪       -n                  \n",
       "  ╱    ⎩      z           for n > 0\n",
       " ╱                                 \n",
       "╱                                  \n",
       "‾‾‾‾‾‾                             \n",
       "n = -∞                             "
      ]
     },
     "execution_count": 61,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TF.rewrite(Piecewise).doit()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## cos"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAK4AAAA5BAMAAABdQQ0mAAAAMFBMVEX///8AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAv3aB7AAAAD3RSTlMAEHaZVCKJq0S772Yyzd3BNagQAAAACXBIWXMAAA7EAAAOxAGVKw4bAAADn0lEQVRYCe1YTWgTQRT+Nsk2227ShhYUBTVYyUWUojaICg0IHlRsFEHFQ4NVUYo/CIJ4cQ9qbQ/SCh6KB61/BysY/EERxUVBRMHUgz+gYhQUsZTW1j+kur7dZHZnswtSOwcPeYd933sz78vbN7MvwwC21C4A1my2TVFA2tdxTm2MxEXxMZ4AsL8eOMRsUVoFop+BkCg+xkO84UHgAbNF6ZCOhdOfSA2i+GyeZfN17Hwes+0yKFdAdAXeGI6MCCRvNnSLTVp//dJPgbxBI8/Y1KEUgwL00HebJNhvw4mDnJG1SZbYaOJANb7aJNTZvHLl1kz/lrTROzeUdnwnfzvYD93owzk/v9Tl493r+KYa/ukUZ8i1eVDX90qUy80enWYjKMYXx+CR0kPSjXBcOcW7Gb7CAK8r4o6162fMMbyoSg9mdK8b7318kE453qjR7xhe1ILqg7rXLee9PvK8cLzSpW+OUUTKJ+1REbYj0GbCuq0LIG1KaLi7Yg6ZAVqVw3vX1McRJkv+aE4haS0o69lsaJxlwSmX53e5ffJZPEw/TuMY7YNqGqrMApmeuPQVN8iSRpBImAG3zUdRgkYvg0zHsIHBoo72Y7n+Acg96EWAfNUpRFKDkMdAiQOb1UxoN+lVpsFkyFuISDcbLOoZFB6hnVOTGTqWJl+VBkn6AXVY0RAFrlVr1nbM8WEted6ycHupJxeno8sP4u2a9GaUBokXoWGE+yn5e1SSGTrOktvFe9NMwCXWO7k8dr6NMdzRrDqAGlVNtg6YBTmTS2OAAo5yQcoYZxTgUsCsFidUX6TOUEJHaAdkC+tWlUFzbLseGUU0nYvhHU3n1y2X4uItGBlom6O5nXIfKtIPdTxTelFJ71fRALRo2KHeP7+oOdkHyvc0RSSdKMm03RLETW6DF8Ymz9sCOdmqKW2zt5FHyVsvve6qdPJp4EIcVF+qPl4VJpvPKP3yP0jJd1zYD1LeYepx4HhQSd8JNJhrzfWdirc227iOwGaReUmYnzDXJ7lNdpWf9zfs29eP21HcJvPvs/bMUuDzP6Q679CSYvOlpn9bQBbv0tJ5Zq49YOgMT1xHf7225BMdp/7yBzquH2tyjmeGyPPZuJIoTy5X4L+pgLr4bn1JMkIuTtTObFU6bJ4ZTxTpxVycrJ6LmpgrYToJiLg42YNORtthph1TxVycXEQyEjab2csiPfEKuDiRh/FxJUvY0iEhFyfKbiSyLl6IuDj5Ayim9euUYCw/AAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} z^{- n} \\cos{\\left (n \\omega_{0} \\right )}$$"
      ],
      "text/plain": [
       "  ∞                 \n",
       " ___                \n",
       " ╲                  \n",
       "  ╲     -n          \n",
       "  ╱    z  ⋅cos(n⋅ω₀)\n",
       " ╱                  \n",
       " ‾‾‾                \n",
       "n = -∞              "
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "radFreq=symbols('omega_0')\n",
    "TF=ZTransDef(cos(radFreq*n), n); TF"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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\n",
      "text/latex": [
       "$$\\sum_{n=-\\infty}^{\\infty} z^{- n} \\cos{\\left (n \\omega_{0} \\right )}$$"
      ],
      "text/plain": [
       "  ∞                 \n",
       " ___                \n",
       " ╲                  \n",
       "  ╲     -n          \n",
       "  ╱    z  ⋅cos(n⋅ω₀)\n",
       " ╱                  \n",
       " ‾‾‾                \n",
       "n = -∞              "
      ]
     },
     "execution_count": 54,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "TF.doit()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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