21-09-27 L05.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 05 - Nyquist-Shannon Theorem\n",
    "Lecture 05 - 27 September 2021\n",
    "\n",
    "## Sampling\n",
    "Given a continuous function $x_A(t)$, it is always possible to generate a discrete sequence by taking $x[n]=x_A(nT)$, where $T$ is the sampling time. A question should naturally arise: how good is it possible to reconstruct a signal $x_{A}(t)$ from its sampled sequence?\n",
    "A first way to do that is to linearly interpolate between points:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle f(x) = sin(x)x^2 - cos(x) log(x+0.1) + 3 \\qquad \\text{in }[0,51] \\text{ with } T=1.7 \\text{ (30 points)}$"
      ],
      "text/plain": [
       "<IPython.core.display.Math object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 1080x216 with 2 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "x = np.linspace(0, 51, 10000)\n",
    "ya= np.sin(x)*x**2 - np.cos(x)*np.log(x+0.1) + 3\n",
    "\n",
    "n = np.linspace(0, 51, 30) # Change here the number of points\n",
    "ys= np.sin(n)*n**2 - np.cos(n)*np.log(n+0.1) + 3\n",
    "\n",
    "\n",
    "fig, ax = plt.subplots(1, 2, figsize=(15, 3))\n",
    "ax[0].plot(x, ya, label=\"analog signal\")\n",
    "ax[0].plot(n, ys, lw=0, marker='.', color='r', label=\"sampling\")\n",
    "\n",
    "ax[1].plot(n, ys, marker='.', linestyle=\"--\", mfc='r', mec='r', label=\"Linear reconstruction\")\n",
    "\n",
    "ax[0].set_xlim(0, 50); ax[1].set_xlim(0, 50);\n",
    "ax[0].set_ylim(-2500, 2500); ax[1].set_ylim(-2500, 2500);\n",
    "ax[0].grid(); ax[1].grid();\n",
    "ax[0].legend(); ax[1].legend()\n",
    "\n",
    "from IPython.display import display, Math, Latex\n",
    "display(Math(r'f(x) = sin(x)x^2 - cos(x) log(x+0.1) + 3 \\qquad \\text{in }[0,51] \\text{ with } T=1.7 \\text{ (30 points)}'))\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "but the results are not great. There exists an optimal way to reconstruct a function from the sampling, but it requires defining some quantities and making some rather general assumptions:\n",
    "\n",
    "### Definitions:\n",
    "- *Sampling period/time*: $T$\n",
    "- *Sampling rate/frequency*: $f_s=T^{-1}$\n",
    "- *Nyquist frequency*: $F_{NY}= \\frac{f_s}{2} \\quad \\omega_{NY} = 2\\pi f_{NY} = \\pi f_s = \\frac{\\pi}{T}$\n",
    "- *Nyquist band:* $f \\in ]-f_{NY}; f_{NY}[ \\quad \\iff \\quad f \\in ]-\\frac{\\pi}{T}; \\frac{\\pi}{T}[$\n",
    "\n",
    "### Assumptions\n",
    "1. $x_A(t)$ is $\\mathcal{L}^2$, that is, it has a Fourier Transform\n",
    "2. $x[n]=x_{A}(nT)$ is _BIBO STABLE_ $\\implies \\Gamma_1 \\in \\text{ROC}\\left\\{ x(z) \\right\\}$\n",
    "\n",
    "Let's start by writing the sampled signal in terms of the Fourier transform of the analog signal:\n",
    "\n",
    "\\begin{align*}\n",
    "x_{s}[n] = x_A(nT)\n",
    "&= \\frac{1}{2\\pi}\\int_{-\\infty}^{+\\infty} \\tilde{x}_A(\\omega)e^{-i\\omega Tn} d\\omega \\\\\n",
    "&= \\frac{1}{2\\pi} \\sum_{k=-\\infty}^{+\\infty} \\int_{-\\frac{\\pi}{T}+\\frac{2\\pi}{T}k}^{\\frac{\\pi}{T}+\\frac{2\\pi}{T}k} \\tilde{x}_A(\\omega)e^{-i\\omega Tn} d\\omega \\qquad \\omega'=\\omega-\\frac{2\\pi}{T}k\\\\\n",
    "&= \\frac{1}{2\\pi} \\sum_{k=-\\infty}^{+\\infty} \\int_{-\\frac{\\pi}{T}}^{\\frac{\\pi}{T}} \\tilde{x}_A\\left(\\omega'+\\frac{2\\pi}{T}k\\right)e^{-i\\left(\\omega'+\\frac{2\\pi}{T}k\\right) Tn} d\\omega'\n",
    "\\qquad e^{\\frac{2\\pi}{T}k Tn}=1 \\\\\n",
    "&= \\frac{1}{2\\pi} \\int_{-\\frac{\\pi}{T}}^{\\frac{\\pi}{T}} \\sum_{k=-\\infty}^{+\\infty} \\tilde{x}_A\\left(\\omega'+\\frac{2\\pi}{T}k\\right)e^{-i\\omega' Tn} d\\omega'\n",
    "\\end{align*}\n",
    "\n",
    "At the same time, the sequence $x_s[n]$ can be expressed via the inverse z-transform around $\\Gamma_1$ clockwise:\n",
    "\n",
    "![Circonference Gamma 1 travelled clockwise](images/21-09-27_Gamma1_clockwise.jpg)\n",
    "\n",
    "\\begin{align*}\n",
    "x_{s}[n] &= \\frac{1}{2\\pi i} \\oint_{\\Gamma \\in ROC} x_s(z) z^{n-1} \\qquad z=e^{-i\\omega t} \\qquad dz=(-iT)e^{-i\\omega T}\\\\\n",
    "&= - \\frac{1}{2\\pi i} \\oint_{-\\frac{\\pi}{T}}^{\\frac{\\pi}{T}} x_s(e^{-i\\omega t}) e^{-i\\omega T (n-1)} (-iT) e^{-i\\omega t} d\\omega \\\\\n",
    "&= \\frac{T}{2\\pi} \\oint_{-\\frac{\\pi}{T}}^{\\frac{\\pi}{T}} x_s(e^{-i\\omega t}) e^{-i\\omega n T} d\\omega\\\\\n",
    "\\end{align*}\n",
    "\n",
    "By composing this expression with the one obtained above it is possible to obtain:\n",
    "```{admonition} Nyquist-Shannon precursor theorem\n",
    "$T x_s\\left(z=e^{-i\\omega T}\\right) = \\sum_{k} \\tilde{x}_A\\left( \\omega+\\frac{2\\pi}{T}k\\right)$ with $\\omega\\in \\left] -\\frac{\\pi}{T}; \\frac{\\pi}{T}\\right[$\n",
    "```\n",
    "\n",
    "**WARNING:** The function on the right must be periodic with period $\\frac{2\\pi}{T}$ due to the nature of the complex exponential. In contrast, the function on the left wasn't assumed to be periodic so the expression can't be true in general. To circumvent this issue, we make a third assumption: \n",
    "\n",
    "\n",
    "3. Assume $x_A(t)$ to be _band limited_ on $\\left] -\\frac{\\pi}{T}; \\frac{\\pi}{T}\\right[$, that is $\\tilde{x}_A(\\omega)=0 \\text{ if } \\left| \\omega\\right| \\geq \\frac{\\pi}{T}$.\n",
    "\n",
    "This assumption allows us to unwrap the summation and force the equality in the precursor theorem:\n",
    "\n",
    "\\begin{align*}\n",
    "    \\require{cancel}\n",
    "    T x_s\\left(z=e^{-i\\omega T}\\right) =\\tilde{x}_A\\left( \\omega\\right) + \\cancelto{0}{ \\tilde{x}_A\\left( \\omega+\\frac{2\\pi}{T}\\right) } +\\ ...\n",
    "\\end{align*}\n",
    "\n",
    "\n",
    "It is now time to go back to the time domain:\n",
    "\\begin{align*}\n",
    "\\require{color}\n",
    "x_A(t) &= \\frac{1}{2\\pi}\\int_{-\\infty}^{+\\infty} \\tilde{x}_A(\\omega)e^{-i\\omega T} d\\omega \\qquad \\text{exploit hypothesis 3.}\\\\\n",
    "&= \\frac{1}{2\\pi}\\int_{-\\frac{\\pi}{T}}^{+\\frac{\\pi}{T}} \\tilde{x}_A(\\omega)e^{-i\\omega T} d\\omega \\qquad \\text{use the precursor theorem}\\\\\n",
    "&= \\frac{1}{2\\pi}\\int_{-\\frac{\\pi}{T}}^{+\\frac{\\pi}{T}} T x_s(z=e^{-i\\omega T}) e^{-i\\omega T} d\\omega\n",
    "= \\frac{T}{2\\pi}\\int_{-\\frac{\\pi}{T}}^{+\\frac{\\pi}{T}} \\sum_{n} x_s[n]e^{i\\omega n T -i\\omega t} d\\omega \\\\\n",
    "&= \\frac{T}{2\\pi}\\sum_{n} x_s[n] \\int_{-\\frac{\\pi}{T}}^{+\\frac{\\pi}{T}} e^{i\\omega n T -i\\omega t} d\\omega \n",
    "= \\frac{T}{2\\pi}\\sum_{n} x_s[n] \\int_{-\\frac{\\pi}{T}}^{+\\frac{\\pi}{T}} e^{i\\omega (n T-t)} d\\omega \\\\\n",
    "&= \\frac{T}{2\\pi}\\sum_{n} x_s[n] \\left[ \\frac{e^{i\\omega(nT-t)}}{i(nT-t)} \\right]^{\\frac{\\pi}{T}}_{-\\frac{\\pi}{T}} = \\cancel{\\frac{T}{2\\pi}} \\sum_{n} x_s[n] \\frac{e^{i(nT-t)\\frac{\\pi}{T}}+e^{-i(nT-t)\\frac{\\pi}{T}}}{i(nT-t) \\textcolor[rgb]{0.75,0.75,0.75}{\\frac{2\\pi}{T}} } \\\\\n",
    "&= \\sum_{n} x_s[n] sinc\\left[ (nT-t)\\frac{\\pi}{T}\\right]\\\\\n",
    "\\end{align*}\n",
    "\n",
    "```{admonition} Nyquist-Shannon theorem in the time domain\n",
    "$x_A(t)=\\sum_{n} x_s[n] sinc\\left[ (nT-t)\\frac{\\pi}{T}\\right]$\n",
    "```\n",
    "\n",
    "The condition $\\tilde{x}_A(\\omega)=0$ if $\\left|\\omega\\right|\\geq \\frac{\\pi}{T}$ or $\\left|f\\right|\\geq \\frac{1}{2T}=f_{NY} \\implies \\max_f\\left\\{ \\left|f\\right|\\right\\}<\\frac{1}{2T}=f_{NY}$.\n",
    "\n",
    "## Aliasing\n",
    "Let's now analyze what happens in the case of **undersampling**, that is, if $f_S < f_{NY}$.\n",
    "\n",
    "### definitions\n",
    "- *Reconstructed Signal*: $x_R(t)=\\sum_{n} x_s[n] sinc\\left[ (nT-t)\\frac{\\pi}{T}\\right]$ with $x_s[n]=x_A(nT)$\n",
    "- *Folding*: $x_F(\\omega)=\\sum_{n}\\tilde{x}_A\\left( \\omega+\\frac{2\\pi}{T}k \\right) \\implies \\tilde{x}_F(\\omega)=T X_s(z=e^{-i\\omega T})$\n",
    "- *Window on Nyquist band*: $W(\\omega)=\\theta\\left(\\omega+\\frac{\\pi}{T}\\right)-\\theta\\left(\\omega-\\frac{\\pi}{T}\\right)$\n",
    "![Window on Nyquist band ](images/21-09-27_window.jpg)\n",
    "\n",
    "- *Alias*: $\\tilde{x}_{alias}(\\omega)=W(\\omega)\\tilde{x}_F(\\omega)$.\n",
    "\n",
    "Properties of the alias function:\n",
    "- It is $\\pi/T$ band limited due to $W(\\omega)$\n",
    "- $\\sum_n \\tilde{x}_{alias}\\left( \\omega+\\frac{2\\pi}{T}k \\right) = \\sum_n \\tilde{x}_{F}\\left( \\omega+\\frac{2\\pi}{T}k \\right)W\\left( \\omega+\\frac{2\\pi}{T}k \\right) = \\tilde{x}_{F}\\left( \\omega\\right)\\cancelto{1}{\\sum_nW\\left( \\omega+\\frac{2\\pi}{T}k \\right)} \\\\\n",
    "\\implies \\tilde{x}_{F}\\left( \\omega\\right) = \\sum_n \\tilde{x}_{alias}\\left( \\omega+\\frac{2\\pi}{T}k \\right) $\n",
    "- If $\\omega \\in \\left] -\\frac{\\pi}{T};\\frac{\\pi}{T} \\right[$ \n",
    "$\\implies$ $\\tilde{x}_{alias}(\\omega)=\\tilde{x}_{F}(\\omega) = T x_s\\left( z=e^{-i\\omega T} \\right)$ \n",
    "$\\implies$ $\\mathcal{FT}^{-1}\\left\\{ \\tilde{x}_{aliasing}(\\omega) \\right\\} = x_R(t)= \\sum_{n} x[n]sinc\\left[ (t-nT)\\frac{\\pi}{T}\\right]$\n",
    "\n",
    "By assuming that $\\tilde{x}_{alias}(\\omega)$ is _band limited_ it is possible to obtain $\\tilde{x}_{alias}(\\omega)=\\tilde{x}_{A}(\\omega) \\implies x_R(t)=x_A(t)$. On the other hand, if $\\tilde{x}_{alias}(\\omega)$ is _NOT_ band limited the reconstruction is not possible:\n",
    "![Comparision between an unersampled signal and a properly samped one](images/21-09-27_over_under_sampling.jpeg)\n",
    "\n",
    "```{warning}\n",
    "DO NOT validate the goodness of a reconstruction on the sampled points as they are _always_ reconstructed perfectly:\n",
    "$\\lim_{t\\rightarrow nT}sinc\\left[(t-nT)\\frac{\\pi}{T}\\right]=1 \\implies x_R(t=nT)=x[n]\\ \\forall n$\n",
    "```"
   ]
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    "### Exercise\n",
    "Take $\\mathcal{Re}\\left\\{e^{i\\omega_0 t}\\right\\} = cos(\\omega_0 t)$ with $\\omega_0=2\\pi f_0$ and sample it first with $T<\\frac{1}{2f_0}$ and then with $T>\\frac{1}{2f_0}$. Reconstruct the two signals and check the differences."
   ]
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