week7.ipynb

{
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  {
   "cell_type": "markdown",
   "source": "# BL40A2030 Wireless Communication Networks\n\n## Week 7: Wireless networks - Outage probability and statistical metrics\n\n## Author: Pedro Nardelli\n\n### References \n\n#### Markdown syntax for writing in the notebook\n\n[Markdown syntax](https://help.github.com/articles/basic-writing-and-formatting-syntax/)\n\n#### Library for numerical calculation in Python\n\n[Numpy](http://www.numpy.org/)\n\n#### Symbolic calculation\n\n[SymPy](https://docs.sympy.org/latest/index.html)\n\n\n#### Plot in Python\n\n[Matplotlib](https://matplotlib.org/tutorials/introductory/pyplot.html)\n\n#### Mathematical functions (like Gamma)\n\n[Math](https://docs.python.org/3/library/math.html)",
   "metadata": {
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    "collapsed": true,
    "cell_id": "00000-80eefad2-e2aa-4b18-9654-7adce1c8d6a0",
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   }
  },
  {
   "cell_type": "code",
   "metadata": {
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    "trusted": true,
    "cell_id": "00001-95a5ff05-88b9-4fb3-a9d1-96075e9caf75",
    "output_cleared": false,
    "source_hash": "14357aa0",
    "execution_millis": 1,
    "execution_start": 1606464024880
   },
   "source": "import numpy as np \nimport matplotlib.pyplot as plt\nimport sympy as sp\nfrom matplotlib import style\nimport math\n#Not to show warning messages (to keep the notebook clean)\nimport warnings\nwarnings.filterwarnings('ignore')",
   "execution_count": 1,
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "source": "# Optimization\n\n## Closed-form with Rayleigh fading (i.i.d. exponential channel gains): $ P_o = 1 - e^{- \\kappa \\lambda \\pi d^2 \\beta^{2/\\alpha}}$\n\n- $\\alpha$ is the path-loss exponent\n- $\\lambda$ is the density of the PPP related to the transmitters\n- $d$ is the TX-RX distance (the reference link, or all links in bipolar networks)\n- $\\beta$ is the SIR threshold that leads to a spectral efficiency of $\\log_2(1+\\beta)$",
   "metadata": {
    "deepnote_cell_type": "markdown",
    "trusted": true,
    "cell_id": "00002-cd036ca7-4521-4810-8055-1baf2c063b25",
    "output_cleared": false
   }
  },
  {
   "cell_type": "markdown",
   "source": "# Performance metrics\n\n### 1) Expected forward progress: $d (1-P_o) =  d  e^{- \\kappa \\lambda \\pi d^2 \\beta^{2/\\alpha}}$\n\n### Optimization in terms of $d$: $\\dfrac{\\mathrm{d}}{\\mathrm{d}d}  \\left(d e^{- \\kappa \\lambda \\pi d^2 \\beta^{2/\\alpha}}\\right) = 0$ \n\nLet's use sympy",
   "metadata": {
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    "cell_id": "00003-e6b6619e-874f-41a3-9f73-5ba45b5b6c7b",
    "output_cleared": false
   }
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00004-6701f24c-3e04-4d96-8002-5d6ded5f2e32",
    "output_cleared": false,
    "source_hash": "6f7bbb96",
    "execution_millis": 0,
    "execution_start": 1606464024881
   },
   "source": "sp.init_printing()\n#Defining variables\nd2, lamb2, beta2, alpha2, kappa2 = sp.symbols('d lambda beta alpha kappa', real = True, positive = True)",
   "execution_count": 2,
   "outputs": []
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00005-1eab783a-a31e-44c4-a1bb-78d53e9c4ccf",
    "output_cleared": false,
    "source_hash": "aa829bac",
    "execution_millis": 138,
    "execution_start": 1606464024907
   },
   "source": "# (1) Defining the variables and functions\n\n#Success probability\np_s= sp.exp(- kappa2* lamb2 * sp.pi * d2**2 * beta2**(2/alpha2))\n\n#EP\nEP = d2 * p_s\n\nEP",
   "execution_count": 3,
   "outputs": [
    {
     "data": {
      "text/plain": "       2       \n       ─       \n       α  2    \n   -π⋅β ⋅d ⋅κ⋅λ\nd⋅ℯ            ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle d e^{- \\pi \\beta^{\\frac{2}{\\alpha}} d^{2} \\kappa \\lambda}$"
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ]
  },
  {
   "cell_type": "code",
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    "cell_id": "00006-aa16aac5-7292-4feb-ba17-6c63271f3698",
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    "source_hash": "f8af8620",
    "execution_millis": 180,
    "execution_start": 1606464025107
   },
   "source": "# (2) Solving the derivative equation -> finding d that optimized EFP\n\nd_opt = sp.solve(sp.diff(EP,d2),d2)\nd_opt[0]",
   "execution_count": 4,
   "outputs": [
    {
     "data": {
      "text/plain": "     -1   \n     ───  \n      α   \n √2⋅β     \n──────────\n2⋅√π⋅√κ⋅√λ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\frac{\\sqrt{2} \\beta^{- \\frac{1}{\\alpha}}}{2 \\sqrt{\\pi} \\sqrt{\\kappa} \\sqrt{\\lambda}}$"
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ]
  },
  {
   "cell_type": "code",
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    "deepnote_cell_type": "code",
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    "cell_id": "00007-03950407-89b5-419e-b10f-6964f8d4e1d6",
    "output_cleared": false,
    "source_hash": "117f6c64",
    "execution_millis": 217,
    "execution_start": 1606464025305
   },
   "source": "#SIR threshold\nbeta=2\n#Path-loss\nalpha=3\n#Constant from Rayleigh fading \nkappa = math.gamma(1 + 2/alpha) * math.gamma(1- 2/alpha)\n#Distance array\nd = np.linspace(0,10,1000)\n####Converting SYMPY to NUMPY -- function of the density lamb2\nd_optimal_np_aux = sp.lambdify(lamb2,sp.N(d_opt[0].subs(beta2,beta).subs(alpha2,alpha).subs(kappa2,kappa)),\"numpy\")\n#\n#Numerical value\nplt.rcParams.update({'font.size': 18})\nplt.figure(figsize=(8,6))\n#Network density\nlamb = 0.01\n#####\nd_optimal_np = d_optimal_np_aux(lamb)\nplt.plot(d, d * np.exp(-kappa * lamb * d**2 * np.pi * beta**(2/alpha)) , '-' , label=r'$\\lambda=0.01$')\nplt.plot(d_optimal_np, d_optimal_np * np.exp(-kappa * lamb * d_optimal_np**2 * np.pi * beta**(2/alpha)) , 'o')\n#Network density\nlamb = 0.015\nd_optimal_np = d_optimal_np_aux(lamb)\nplt.plot(d, d * np.exp(-kappa * lamb * d**2 * np.pi * beta**(2/alpha)) , ':' , label=r'$\\lambda=0.015$')\nplt.plot(d_optimal_np, d_optimal_np * np.exp(-kappa * lamb * d_optimal_np**2 * np.pi * beta**(2/alpha)) , 'o')\n#Network density\nlamb = 0.25\nd_optimal_np = d_optimal_np_aux(lamb)\nplt.plot(d, d * np.exp(-kappa * lamb * d**2 * np.pi * beta**(2/alpha)) ,'--' , label=r'$\\lambda=0.25$')\nplt.plot(d_optimal_np, d_optimal_np * np.exp(-kappa * lamb * d_optimal_np**2 * np.pi * beta**(2/alpha)) , 'o')\n#######\nplt.xlabel(\"Distance\")\nplt.ylabel(\"EFP [m]\")\nplt.grid()\nplt.legend()\nplt.show()",
   "execution_count": 5,
   "outputs": [
    {
     "data": {
      "text/plain": "<Figure size 576x432 with 1 Axes>",
      "image/png": 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\n"
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ]
  },
  {
   "cell_type": "markdown",
   "source": "### 2) Information efficiency: $d \\times  e^{- \\kappa \\lambda \\pi d^2 \\beta^{2/\\alpha}} \\times \\log_2(1+\\beta)$\n\n- We assume Shannon to related the spectral efficiency in [bits/s/Hz] with the SIR threshold.\n- Optimization in terms of $\\beta$",
   "metadata": {
    "deepnote_cell_type": "markdown",
    "cell_id": "00008-9fab6b54-deb8-4f01-9553-5d508fdec51c",
    "output_cleared": false
   }
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00009-b7488552-0938-4b92-a6f0-e4ecc99c118d",
    "output_cleared": false,
    "source_hash": "c646d225",
    "execution_millis": 157,
    "execution_start": 1606464025547
   },
   "source": "# (1) Defining the variables and functions\n\n#EP\nIE = d2 * p_s * sp.log(1+beta2)/sp.log(2)\n\nIE",
   "execution_count": 6,
   "outputs": [
    {
     "data": {
      "text/plain": "       2                  \n       ─                  \n       α  2               \n   -π⋅β ⋅d ⋅κ⋅λ           \nd⋅ℯ            ⋅log(β + 1)\n──────────────────────────\n          log(2)          ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\frac{d e^{- \\pi \\beta^{\\frac{2}{\\alpha}} d^{2} \\kappa \\lambda} \\log{\\left(\\beta + 1 \\right)}}{\\log{\\left(2 \\right)}}$"
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00010-e4d5f341-814d-4ad9-ab9c-a47d5c8948ae",
    "output_cleared": false,
    "source_hash": "1433bf90",
    "execution_millis": 1219,
    "execution_start": 1606464025774
   },
   "source": "# (2) Solving the derivative equation -> finding d that optimized EFP\n\nbeta_opt = sp.solve(sp.diff(IE,beta2),beta2)\nbeta_opt[0]",
   "execution_count": 7,
   "outputs": [
    {
     "ename": "NotImplementedError",
     "evalue": "multiple generators [beta, beta**(1/alpha), exp(pi*beta**(2/alpha)*d**2*kappa*lambda), log(beta + 1)]\nNo algorithms are implemented to solve equation d*exp(-pi*beta**(2/alpha)*d**2*kappa*lambda)/((beta + 1)*log(2)) - 2*pi*beta**(2/alpha)*d**3*kappa*lambda*exp(-pi*beta**(2/alpha)*d**2*kappa*lambda)*log(beta + 1)/(alpha*beta*log(2))",
     "output_type": "error",
     "traceback": [
      "\u001B[1;31m---------------------------------------------------------------------------\u001B[0m",
      "\u001B[1;31mNotImplementedError\u001B[0m                       Traceback (most recent call last)",
      "\u001B[1;32m<ipython-input-7-f6b0db904d87>\u001B[0m in \u001B[0;36m<module>\u001B[1;34m\u001B[0m\n\u001B[0;32m      1\u001B[0m \u001B[1;31m# (2) Solving the derivative equation -> finding d that optimized EFP\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0;32m      2\u001B[0m \u001B[1;33m\u001B[0m\u001B[0m\n\u001B[1;32m----> 3\u001B[1;33m \u001B[0mbeta_opt\u001B[0m \u001B[1;33m=\u001B[0m \u001B[0msp\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0msolve\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0msp\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0mdiff\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0mIE\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0mbeta2\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0mbeta2\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0m\u001B[0;32m      4\u001B[0m \u001B[0mbeta_opt\u001B[0m\u001B[1;33m[\u001B[0m\u001B[1;36m0\u001B[0m\u001B[1;33m]\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0;32m      5\u001B[0m \u001B[1;33m\u001B[0m\u001B[0m\n",
      "\u001B[1;32mC:\\ProgramData\\Anaconda3\\envs\\wireless-net\\lib\\site-packages\\sympy\\solvers\\solvers.py\u001B[0m in \u001B[0;36msolve\u001B[1;34m(f, *symbols, **flags)\u001B[0m\n\u001B[0;32m   1095\u001B[0m     \u001B[1;31m###########################################################################\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0;32m   1096\u001B[0m     \u001B[1;32mif\u001B[0m \u001B[0mbare_f\u001B[0m\u001B[1;33m:\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[1;32m-> 1097\u001B[1;33m         \u001B[0msolution\u001B[0m \u001B[1;33m=\u001B[0m \u001B[0m_solve\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0mf\u001B[0m\u001B[1;33m[\u001B[0m\u001B[1;36m0\u001B[0m\u001B[1;33m]\u001B[0m\u001B[1;33m,\u001B[0m \u001B[1;33m*\u001B[0m\u001B[0msymbols\u001B[0m\u001B[1;33m,\u001B[0m \u001B[1;33m**\u001B[0m\u001B[0mflags\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0m\u001B[0;32m   1098\u001B[0m     \u001B[1;32melse\u001B[0m\u001B[1;33m:\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0;32m   1099\u001B[0m         \u001B[0msolution\u001B[0m \u001B[1;33m=\u001B[0m \u001B[0m_solve_system\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0mf\u001B[0m\u001B[1;33m,\u001B[0m \u001B[0msymbols\u001B[0m\u001B[1;33m,\u001B[0m \u001B[1;33m**\u001B[0m\u001B[0mflags\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n",
      "\u001B[1;32mC:\\ProgramData\\Anaconda3\\envs\\wireless-net\\lib\\site-packages\\sympy\\solvers\\solvers.py\u001B[0m in \u001B[0;36m_solve\u001B[1;34m(f, *symbols, **flags)\u001B[0m\n\u001B[0;32m   1714\u001B[0m \u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0;32m   1715\u001B[0m     \u001B[1;32mif\u001B[0m \u001B[0mresult\u001B[0m \u001B[1;32mis\u001B[0m \u001B[1;32mFalse\u001B[0m\u001B[1;33m:\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[1;32m-> 1716\u001B[1;33m         \u001B[1;32mraise\u001B[0m \u001B[0mNotImplementedError\u001B[0m\u001B[1;33m(\u001B[0m\u001B[1;34m'\\n'\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0mjoin\u001B[0m\u001B[1;33m(\u001B[0m\u001B[1;33m[\u001B[0m\u001B[0mmsg\u001B[0m\u001B[1;33m,\u001B[0m \u001B[0mnot_impl_msg\u001B[0m \u001B[1;33m%\u001B[0m \u001B[0mf\u001B[0m\u001B[1;33m]\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0m\u001B[0;32m   1717\u001B[0m \u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0;32m   1718\u001B[0m     \u001B[1;32mif\u001B[0m \u001B[0mflags\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0mget\u001B[0m\u001B[1;33m(\u001B[0m\u001B[1;34m'simplify'\u001B[0m\u001B[1;33m,\u001B[0m \u001B[1;32mTrue\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m:\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n",
      "\u001B[1;31mNotImplementedError\u001B[0m: multiple generators [beta, beta**(1/alpha), exp(pi*beta**(2/alpha)*d**2*kappa*lambda), log(beta + 1)]\nNo algorithms are implemented to solve equation d*exp(-pi*beta**(2/alpha)*d**2*kappa*lambda)/((beta + 1)*log(2)) - 2*pi*beta**(2/alpha)*d**3*kappa*lambda*exp(-pi*beta**(2/alpha)*d**2*kappa*lambda)*log(beta + 1)/(alpha*beta*log(2))"
     ]
    }
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00011-e6717379-343b-412b-986b-20edda6d051f",
    "output_cleared": false,
    "source_hash": "e51d7fd",
    "execution_start": 1606464196532,
    "execution_millis": 275
   },
   "source": "#Conditions of existence of solution -> https://docs.sympy.org/latest/tutorial/solvers.html\nbeta_opt = sp.solveset(sp.diff(IE, beta2),beta2)\nbeta_opt",
   "execution_count": null,
   "outputs": []
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00012-7b462089-9da8-43d0-8368-53c534a444b8",
    "output_cleared": false,
    "source_hash": "f00da87d",
    "execution_start": 1606464197580,
    "execution_millis": 340
   },
   "source": "#Checking the derivative -- transcendental equation\nIE_dif = sp.simplify(sp.diff(IE,beta2))\nIE_dif",
   "execution_count": null,
   "outputs": []
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00013-79e93d56-8a15-41ed-8406-426e0d31ecc8",
    "output_cleared": false,
    "source_hash": "77896149",
    "execution_start": 1606464198615,
    "execution_millis": 7170
   },
   "source": "#Numerical solution 1\nd=1\nalpha=3\nlamb=0.1\nsp.solve(IE_dif.subs(alpha2,alpha).subs(kappa2,kappa).subs(lamb2,lamb).subs(d2,d),beta2,1)",
   "execution_count": null,
   "outputs": []
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00014-9fd90e02-f1d6-45df-865c-d183e9c4dd42",
    "output_cleared": false,
    "source_hash": "32ecc848",
    "execution_start": 1606464216628,
    "execution_millis": 1
   },
   "source": "#Numerical solution 2\nd=1\nalpha=3\nlamb=0.1\n#### NOte that the last input in the nsolve is where the solution should be around; many times you need to figure out this yourself.\nsp.nsolve(IE_dif.subs(alpha2,alpha).subs(kappa2,kappa).subs(lamb2,lamb).subs(d2,d),beta2,0.1)",
   "execution_count": 10,
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'IE_dif' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001B[1;31m---------------------------------------------------------------------------\u001B[0m",
      "\u001B[1;31mNameError\u001B[0m                                 Traceback (most recent call last)",
      "\u001B[1;32m<ipython-input-10-86edb48b197c>\u001B[0m in \u001B[0;36m<module>\u001B[1;34m\u001B[0m\n\u001B[0;32m      4\u001B[0m \u001B[0mlamb\u001B[0m\u001B[1;33m=\u001B[0m\u001B[1;36m0.1\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0;32m      5\u001B[0m \u001B[1;31m#### NOte that the last input in the nsolve is where the solution should be around; many times you need to figure out this yourself.\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[1;32m----> 6\u001B[1;33m \u001B[0msp\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0mnsolve\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0mIE_dif\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0msubs\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0malpha2\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0malpha\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0msubs\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0mkappa2\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0mkappa\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0msubs\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0mlamb2\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0mlamb\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0msubs\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0md2\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0md\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0mbeta2\u001B[0m\u001B[1;33m,\u001B[0m\u001B[1;36m0.1\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0m\u001B[0;32m      7\u001B[0m \u001B[1;33m\u001B[0m\u001B[0m\n",
      "\u001B[1;31mNameError\u001B[0m: name 'IE_dif' is not defined"
     ]
    }
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00015-6f14177a-109d-43e8-b4b3-a5073e8b7b13",
    "output_cleared": false,
    "source_hash": "6fb57d67",
    "execution_start": 1606464219247,
    "execution_millis": 341
   },
   "source": "#Path-loss\nalpha=3\n#Constant from Rayleigh fading \nkappa = math.gamma(1 + 2/alpha) * math.gamma(1- 2/alpha)\n#Network density\nlamb = 0.05\n#SIR threshold array\nbeta = np.linspace(0,8,300)\n#####\nplt.rcParams.update({'font.size': 18})\nplt.figure(figsize=(14,8))\n#distance\nd=1\nbeta_opt_aux = sp.nsolve(IE_dif.subs(alpha2,alpha).subs(kappa2,kappa).subs(lamb2,lamb).subs(d2,d),beta2,0.1)\nbeta_opt_np = sp.lambdify([],beta_opt_aux,\"numpy\")\nplt.plot(beta, d * np.exp(-kappa * lamb * d**2 * np.pi * beta**(2/alpha)) * np.log2(1+beta) , '-' , label=r'$d=1$')\nplt.plot(beta_opt_np(), d * np.exp(-kappa * lamb * d**2 * np.pi * beta_opt_np()**(2/alpha))* np.log2(1+beta_opt_np()) ,'o')\n#distance\nd=2\nbeta_opt_aux = sp.nsolve(IE_dif.subs(alpha2,alpha).subs(kappa2,kappa).subs(lamb2,lamb).subs(d2,d),beta2,0.1)\nbeta_opt_np = sp.lambdify([],beta_opt_aux,\"numpy\")\nplt.plot(beta, d * np.exp(-kappa * lamb * d**2 * np.pi * beta**(2/alpha)) * np.log2(1+beta), ':' , label=r'$d=2$')\nplt.plot(beta_opt_np(), d * np.exp(-kappa * lamb * d**2 * np.pi * beta_opt_np()**(2/alpha))* np.log2(1+beta_opt_np()) ,'o')\n######\nd=3\nbeta_opt_aux = sp.nsolve(IE_dif.subs(alpha2,alpha).subs(kappa2,kappa).subs(lamb2,lamb).subs(d2,d),beta2,0.1)\nbeta_opt_np = sp.lambdify([],beta_opt_aux,\"numpy\")\nplt.plot(beta, d * np.exp(-kappa * lamb * d**2 * np.pi * beta**(2/alpha))* np.log2(1+beta) ,'--' , label=r'$d=3$')\nplt.plot(beta_opt_np(), d * np.exp(-kappa * lamb * d**2 * np.pi * beta_opt_np()**(2/alpha))* np.log2(1+beta_opt_np()) ,'o')\n#######\nplt.xlabel(\"SIR threshold\")\nplt.ylabel(\"IE [bits-m/s/Hz]\")\nplt.grid()\nplt.legend()\nplt.show()",
   "execution_count": 11,
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'IE_dif' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001B[1;31m---------------------------------------------------------------------------\u001B[0m",
      "\u001B[1;31mNameError\u001B[0m                                 Traceback (most recent call last)",
      "\u001B[1;32m<ipython-input-11-dd41fc2e2a9b>\u001B[0m in \u001B[0;36m<module>\u001B[1;34m\u001B[0m\n\u001B[0;32m     12\u001B[0m \u001B[1;31m#distance\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0;32m     13\u001B[0m \u001B[0md\u001B[0m\u001B[1;33m=\u001B[0m\u001B[1;36m1\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[1;32m---> 14\u001B[1;33m \u001B[0mbeta_opt_aux\u001B[0m \u001B[1;33m=\u001B[0m \u001B[0msp\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0mnsolve\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0mIE_dif\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0msubs\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0malpha2\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0malpha\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0msubs\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0mkappa2\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0mkappa\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0msubs\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0mlamb2\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0mlamb\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0msubs\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0md2\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0md\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0mbeta2\u001B[0m\u001B[1;33m,\u001B[0m\u001B[1;36m0.1\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0m\u001B[0;32m     15\u001B[0m \u001B[0mbeta_opt_np\u001B[0m \u001B[1;33m=\u001B[0m \u001B[0msp\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0mlambdify\u001B[0m\u001B[1;33m(\u001B[0m\u001B[1;33m[\u001B[0m\u001B[1;33m]\u001B[0m\u001B[1;33m,\u001B[0m\u001B[0mbeta_opt_aux\u001B[0m\u001B[1;33m,\u001B[0m\u001B[1;34m\"numpy\"\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n\u001B[0;32m     16\u001B[0m \u001B[0mplt\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0mplot\u001B[0m\u001B[1;33m(\u001B[0m\u001B[0mbeta\u001B[0m\u001B[1;33m,\u001B[0m \u001B[0md\u001B[0m \u001B[1;33m*\u001B[0m \u001B[0mnp\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0mexp\u001B[0m\u001B[1;33m(\u001B[0m\u001B[1;33m-\u001B[0m\u001B[0mkappa\u001B[0m \u001B[1;33m*\u001B[0m \u001B[0mlamb\u001B[0m \u001B[1;33m*\u001B[0m \u001B[0md\u001B[0m\u001B[1;33m**\u001B[0m\u001B[1;36m2\u001B[0m \u001B[1;33m*\u001B[0m \u001B[0mnp\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0mpi\u001B[0m \u001B[1;33m*\u001B[0m \u001B[0mbeta\u001B[0m\u001B[1;33m**\u001B[0m\u001B[1;33m(\u001B[0m\u001B[1;36m2\u001B[0m\u001B[1;33m/\u001B[0m\u001B[0malpha\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m)\u001B[0m \u001B[1;33m*\u001B[0m \u001B[0mnp\u001B[0m\u001B[1;33m.\u001B[0m\u001B[0mlog2\u001B[0m\u001B[1;33m(\u001B[0m\u001B[1;36m1\u001B[0m\u001B[1;33m+\u001B[0m\u001B[0mbeta\u001B[0m\u001B[1;33m)\u001B[0m \u001B[1;33m,\u001B[0m \u001B[1;34m'-'\u001B[0m \u001B[1;33m,\u001B[0m \u001B[0mlabel\u001B[0m\u001B[1;33m=\u001B[0m\u001B[1;34mr'$d=1$'\u001B[0m\u001B[1;33m)\u001B[0m\u001B[1;33m\u001B[0m\u001B[1;33m\u001B[0m\u001B[0m\n",
      "\u001B[1;31mNameError\u001B[0m: name 'IE_dif' is not defined"
     ]
    },
    {
     "data": {
      "text/plain": "<Figure size 1008x576 with 0 Axes>"
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ]
  },
  {
   "cell_type": "markdown",
   "source": "### 3) Spatial throughput/Area spectral efficiency (for bipolar networks): $\\lambda \\times  e^{- \\kappa \\lambda \\pi d^2 \\beta^{2/\\alpha}} \\times \\log_2(1+\\beta)$\n\n- Bipolar network\n- Optimization in terms of $\\lambda$",
   "metadata": {
    "deepnote_cell_type": "markdown",
    "cell_id": "00016-d31ff9e2-3556-483c-9282-4fa9b05df063",
    "output_cleared": false
   }
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00017-fc32829a-d3ce-4bf2-8b9b-9684aef629aa",
    "output_cleared": false,
    "source_hash": "b27c352c",
    "execution_start": 1606464223660,
    "execution_millis": 129
   },
   "source": "# (1) Defining the variables and functions\n\n#EP\nST = lamb2 * p_s * sp.log(1+beta2)/sp.log(2)\n\nST",
   "execution_count": 8,
   "outputs": [
    {
     "data": {
      "text/plain": "       2                  \n       ─                  \n       α  2               \n   -π⋅β ⋅d ⋅κ⋅λ           \nλ⋅ℯ            ⋅log(β + 1)\n──────────────────────────\n          log(2)          ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\frac{\\lambda e^{- \\pi \\beta^{\\frac{2}{\\alpha}} d^{2} \\kappa \\lambda} \\log{\\left(\\beta + 1 \\right)}}{\\log{\\left(2 \\right)}}$"
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00018-18e0b2bf-b9e5-4f0b-8bc4-1daf682e9ed8",
    "output_cleared": false,
    "source_hash": "be4b227a",
    "execution_start": 1606464225265,
    "execution_millis": 164
   },
   "source": "# (2) Solving the derivative equation -> finding d that optimized EFP\n\nlamb_opt = sp.solve(sp.diff(ST,lamb2),lamb2)\nlamb_opt[0]",
   "execution_count": 9,
   "outputs": [
    {
     "data": {
      "text/plain": "  -2  \n  ─── \n   α  \n β    \n──────\n   2  \nπ⋅d ⋅κ",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\frac{\\beta^{- \\frac{2}{\\alpha}}}{\\pi d^{2} \\kappa}$"
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ]
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00019-3d36d45b-f69d-4d11-8739-5e8e9d681453",
    "output_cleared": false,
    "source_hash": "22b066c8",
    "execution_start": 1606464227678,
    "execution_millis": 63
   },
   "source": "lamb_optimal_np_aux = sp.lambdify([],sp.N(lamb_opt[0].subs(beta2,beta).subs(alpha2,alpha).subs(kappa2,kappa).subs(d2,d)),\"numpy\")\nlamb_optimal_np = lamb_optimal_np_aux()\nlamb_optimal_np ",
   "execution_count": null,
   "outputs": []
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00020-37e45e48-c62d-49cd-a7c5-f9adf1bbfd92",
    "output_cleared": false,
    "source_hash": "c439d223",
    "execution_start": 1606464229314,
    "execution_millis": 315
   },
   "source": "#Path-loss\nalpha=3\n#Constant from Rayleigh fading \nkappa = math.gamma(1 + 2/alpha) * math.gamma(1- 2/alpha)\n#Network density\nlamb = np.linspace(0,0.4,200)\n#####\n#####\nplt.rcParams.update({'font.size': 18})\nplt.figure(figsize=(14,8))\n###########\n#distance\nd=1\n#SIR th\nbeta=1\n##\n####Converting SYMPY to NUMPY -- function of the density lamb2\nlamb_optimal_np_aux = sp.lambdify([],sp.N(lamb_opt[0].subs(beta2,beta).subs(alpha2,alpha).subs(kappa2,kappa).subs(d2,d)),\"numpy\")\nlamb_optimal_np = lamb_optimal_np_aux()\nplt.plot(lamb, lamb * d * np.exp(-kappa * lamb * d**2 * np.pi * beta**(2/alpha)) * np.log2(1+beta) , '-' , label=r'$d=1,\\beta=1$')\nplt.plot(lamb_optimal_np, lamb_optimal_np * d * np.exp(-kappa * lamb_optimal_np * d**2 * np.pi * beta**(2/alpha)) * np.log2(1+beta) , 'o' )\n#distance\nd=1\n#SIR th\nbeta=3\n####Converting SYMPY to NUMPY -- function of the density lamb2\nlamb_optimal_np_aux = sp.lambdify([],sp.N(lamb_opt[0].subs(beta2,beta).subs(alpha2,alpha).subs(kappa2,kappa).subs(d2,d)),\"numpy\")\nlamb_optimal_np = lamb_optimal_np_aux()\nplt.plot(lamb, lamb * d * np.exp(-kappa * lamb * d**2 * np.pi * beta**(2/alpha)) * np.log2(1+beta), ':' , label=r'$d=1,\\beta=3$')\nplt.plot(lamb_optimal_np, lamb_optimal_np * d * np.exp(-kappa * lamb_optimal_np * d**2 * np.pi * beta**(2/alpha)) * np.log2(1+beta) , 'o' )\n###########\n#distance\nd=3\n#SIR th\nbeta=1\n####Converting SYMPY to NUMPY -- function of the density lamb2\nlamb_optimal_np_aux = sp.lambdify([],sp.N(lamb_opt[0].subs(beta2,beta).subs(alpha2,alpha).subs(kappa2,kappa).subs(d2,d)),\"numpy\")\nlamb_optimal_np = lamb_optimal_np_aux()\nplt.plot(lamb, lamb * d * np.exp(-kappa * lamb * d**2 * np.pi * beta**(2/alpha))* np.log2(1+beta) ,'--' , label=r'$d=3,\\beta=1$')\nplt.plot(lamb_optimal_np, lamb_optimal_np * d * np.exp(-kappa * lamb_optimal_np * d**2 * np.pi * beta**(2/alpha)) * np.log2(1+beta) , 'o' )\n###########\n#distance\nd=3\n#SIR th\nbeta=3\n####Converting SYMPY to NUMPY -- function of the density lamb2\nlamb_optimal_np_aux = sp.lambdify([],sp.N(lamb_opt[0].subs(beta2,beta).subs(alpha2,alpha).subs(kappa2,kappa).subs(d2,d)),\"numpy\")\nlamb_optimal_np = lamb_optimal_np_aux()\nplt.plot(lamb, lamb *  d * np.exp(-kappa * lamb * d**2 * np.pi * beta**(2/alpha))* np.log2(1+beta) ,'-.' ,label=r'$d=3,\\beta=3$')\nplt.plot(lamb_optimal_np, lamb_optimal_np * d * np.exp(-kappa * lamb_optimal_np * d**2 * np.pi * beta**(2/alpha)) * np.log2(1+beta) , 'o' )\n###########\n###########\n#distance\nd=2\n#SIR th\nbeta=2\n####Converting SYMPY to NUMPY -- function of the density lamb2\nlamb_optimal_np_aux = sp.lambdify([],sp.N(lamb_opt[0].subs(beta2,beta).subs(alpha2,alpha).subs(kappa2,kappa).subs(d2,d)),\"numpy\")\nlamb_optimal_np = lamb_optimal_np_aux()\nplt.plot(lamb, lamb *  d * np.exp(-kappa * lamb * d**2 * np.pi * beta**(2/alpha))* np.log2(1+beta) ,'-' ,label=r'$d=2,\\beta=2$')\nplt.plot(lamb_optimal_np, lamb_optimal_np * d * np.exp(-kappa * lamb_optimal_np * d**2 * np.pi * beta**(2/alpha)) * np.log2(1+beta) , 'o' )\n###########\nplt.xlabel(\"Network density\")\nplt.ylabel(\"ST [bits/s/Hz/m^2]\")\nplt.grid()\nplt.legend()\nplt.show()",
   "execution_count": null,
   "outputs": []
  },
  {
   "cell_type": "markdown",
   "source": "### Try yourself with all metrics with different values",
   "metadata": {
    "deepnote_cell_type": "markdown",
    "cell_id": "00021-c07ca4ec-1a3f-4354-81ae-f2549d1296d1",
    "output_cleared": false
   }
  },
  {
   "cell_type": "code",
   "metadata": {
    "deepnote_cell_type": "code",
    "trusted": true,
    "cell_id": "00022-587f4874-1bbb-43c3-bd3c-8708a08ff7d6",
    "output_cleared": false,
    "source_hash": "b623e53d"
   },
   "source": "",
   "execution_count": null,
   "outputs": []
  }
 ],
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