{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
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     "end_time": "2021-09-27T19:15:08.678463Z",
     "start_time": "2021-09-27T19:15:07.521881Z"
    },
    "tags": [
     "remove-cell"
    ]
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from scipy import special"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Lecture 2: Noise processes and measurement sensitivity\n",
    "\n",
    "```{admonition} Expected prior knowledge\n",
    ":class: tip\n",
    "Before the start of this lecture, you should be able to:\n",
    "\n",
    "- write down the relation between the autocorrelation function and the power spectral density\n",
    "- describe the power spectral density of white and 1/f noise processes \n",
    "```\n",
    "\n",
    "```{admonition} Learning goals\n",
    ":class: important\n",
    "After this lecture you will be able to:\n",
    "\n",
    "- describe the Poissonian and Gaussian probability distributions and argue when they arise\n",
    "- relate the noise power spectral density of a sensor to its ability to detect a small signal\n",
    "```\n",
    "\n",
    "Where does noise come from?\n",
    "\n",
    "In general, noise is caused by processes we don't know about, or at least don't \n",
    "know enough about to predict. A good example of this is the Brownian motion\n",
    "of a particle in a liquid, which for the right type of particle, one can even\n",
    "see in a microscope.\n",
    "\n",
    "Due to random collisions with molecules in the liquid, the particle experiences\n",
    "a randomly fluctuating force: it experiences \"force noise\", and undergoes random\n",
    "motion in time.\n",
    "\n",
    "[<div align=\"center\"><img src=\"figures/2/drift_force_sketch.PNG\" width=\"300\"/></div>](drift_force_sketch.png)\n",
    "\n",
    "What makes these collision events \"random\"? Newton's laws certainly are not random.\n",
    "The events are random because we do not have the information needed to predict them: \n",
    "we would need to know the positions and velocities of all the molecules\n",
    "in the liquid. If there are many molecules (large N) and we cannot measure\n",
    "fast enough to observe the motion of the particle in-between collisions then\n",
    "the particle's motion will look random. If we furthermore neglect the time correlations \n",
    "of the motion of the molecules in the liquid, we can describe the random force acting on our particle \n",
    "by the autocorrelation function\n",
    "\n",
    "$$\n",
    "R_{FF}(\\tau) \\propto \\delta(\\tau) \\rightarrow S_{FF}(\\omega) = \\, \\text{Constant (white noise)}\n",
    "$$\n",
    "\n",
    "For such random behavior, what is the probability that if we look at a certain \n",
    "time, we will find the particle at a given position? In other words, what is the probability\n",
    "density function $p(x)$?\n",
    "\n",
    "In the previous lecture, we drew some cartoons of such distributions with a shape\n",
    "that looked like a Bell curve, corresponding to a Gaussian probability \n",
    "distribution (also called a \"normal\" distribution):\n",
    "\n",
    "$$\n",
    "p(x) = \\frac{1}{\\sqrt{2\\pi \\sigma^2}} e^{-\\frac{(x-\\mu)^2}{2\\sigma^2}}\n",
    "$$\n",
    "\n",
    "<!--\n",
    "[<div align=\"center\"><img src=\"figures/2/gaussian_fwhm.png\" width=\"300\"/></div>](gaussian_fwhm.png)\n",
    "--->\n",
    "\n",
    "This is a very commonly observed distribution function (hence the name \"normal\").\n",
    "The reason it is so common is due to the Central Limit Theorem. \n",
    "\n",
    "```{admonition} Central Limit Theorem\n",
    "This theorem tell us that if we\n",
    "1. collect a large number of independent (uncorrelated) observations/measurements of a statistical quantity  \n",
    "2. calculate the average of these observations\n",
    "3. repeat steps 1-2 many times to create a collection of averages \n",
    "```\n",
    "\n",
    "then the distribution of the averages will always converge to a Gaussian \n",
    "(normal) distribution in the large $N$ limit. More generally, the central limit theorem \n",
    "tells us that the probability distribution of the sum of many stochastic variables will converge to a Gaussian. \n",
    "\n",
    "**Example: 1D random walk**\n",
    "\n",
    "We consider a particle on a discrete 1D grid. We repeatedly flip a coin and move\n",
    "the particle left or right by one step depending on the outcome of the coin\n",
    "flip. This generates a simplified version of the Brownian motion of a particle. \n",
    "The final position of the particle will be given by the net number of steps in a\n",
    "particular direction. \n",
    "\n",
    "The probability of finding the particle at a certain final position after $N$ coin flips \n",
    "is described by a Binomial distribution. When $N$ is large, this \n",
    "distribution converges to a Gaussian as described by the central limit theorem.\n",
    "\n",
    "Note that this is an \"unconfined\" random walk: the longer we wait (i.e., the larger the number of coin flips $N$),\n",
    " the larger the standard deviation of the distribution will become.\n",
    "We will later also look at the case of a harmonic oscillator. Here the \"walk\"\n",
    "of the particle is confined by the walls of the potential: the distribution\n",
    "is still Gaussian but now static in time with a width that depends on the \n",
    "average (thermal) energy."
   ]
  },
  {
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     "start_time": "2021-09-27T19:15:08.681073Z"
    },
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ts = np.linspace(-3,3,100)\n",
    "def gaussian(ts, mu, sigma):\n",
    "    return 1/(sigma * np.sqrt(2 * np.pi)) * np.exp( - (ts - mu)**2 / (2 * sigma**2))\n",
    "plt.plot(ts, 15*gaussian(ts, 0, 1), label=\"Walker position probability distribution\")\n",
    "plt.plot(ts, [t**2 for t in ts], label=\"Harmonic confining potential\")\n",
    "plt.legend()\n",
    "plt.xlabel(\"Position\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Example: Poisson disribution**\n",
    "\n",
    "If events occur randomly in time at a constant average rate $\\lambda$, then the Poisson distribution describes the probability of observing a particular number of events within a time interval $T$. On average, we expect\n",
    "to observe $\\mu = \\lambda T$ events within this time interval. The probability for observing $n$ events within $T$ is given by\n",
    "\n",
    "$$\n",
    "P(n) =  \\frac{\\mu^n}{n!} e^{-\\mu} \n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "ExecuteTime": {
     "end_time": "2021-09-27T19:15:09.318191Z",
     "start_time": "2021-09-27T19:15:08.850508Z"
    },
    "lines_to_next_cell": 0,
    "tags": [
     "hide-input"
    ]
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ns = np.linspace(0,40,41)\n",
    "def poisson(ns, mu):\n",
    "    return (np.power(mu, ns) / special.factorial(ns))*np.exp(-mu)\n",
    "plt.bar(ns, poisson(ns, 5), label=\"mu: \" + str(5))\n",
    "plt.bar(ns, poisson(ns, 10), label=\"mu: \" + str(10))\n",
    "plt.bar(ns, poisson(ns, 20), label=\"mu: \" + str(20))\n",
    "#plt.plot(ns, poisson(ns, 5), label=\"mu: \" + str(5))\n",
    "#plt.plot(ns, poisson(ns, 10), label=\"mu: \" + str(10))\n",
    "#plt.plot(ns, poisson(ns, 20), label=\"mu: \" + str(20))\n",
    "plt.title(\"Poisson Distribution\")\n",
    "plt.ylabel(\"Probability\")\n",
    "plt.xlabel(\"Number of events\")\n",
    "plt.legend()\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An example of Poissonian noise is the \"shot noise\" of a laser: a laser emits photons with a certain average rate $\\lambda$ that is proportional to the average laser power. Because the individual photon emission events occur randomly in time, the probability of emitting $n$ photons in a time $T$ follows a Poisson distribution parametrized by $\\mu=\\lambda T$. The standard deviation of such a Poisson distribution is $\\sqrt{\\mu}$. \n",
    "\n",
    "```{admonition} Question\n",
    ":class: question\n",
    "How can we calculate the fluctuations in the laser power caused by the probabalistic nature of the photon emission events?\n",
    "```\n",
    "1. We know that the standard deviation of the number of emitted photons is $\\sqrt{\\mu}$.\n",
    "2. We also know that the average laser power is given by the average photon emission rate times the energy of a photon $\\hbar\\omega$:\n",
    "\n",
    "$$\n",
    "P = \\mu \\frac{\\hbar \\omega_0}{T}\n",
    "$$\n",
    "\n",
    "3. The fluctuations in $P$ are due to the fluctations in $\\mu$:\n",
    "\n",
    "$$\n",
    "\\sigma_P = \\sqrt{\\mu}\\frac{\\hbar \\omega_0}{T} = \\sqrt{\\frac{P\\hbar \\omega}{T}} \n",
    "$$\n",
    "\n",
    "4. To determine the laser's power spectral density, we measure the laser power repeatedly, each time for a time $T$. The measured value will fluctuate with a standard deviation $\\sigma_P$.\n",
    "5. Our measurement bandwidth is given by the sample frequency $\\frac{1}{T}$. We can thus calculate the power spectral density of the laser power:\n",
    "\n",
    "$$\n",
    "S_{pp} = \\frac{P \\hbar \\omega_0}{T} \\cdot \\frac{T}{1} = \n",
    "P \\hbar \\omega_0 \\quad \\big( \\text{Watts}^2 / \\text{Hz} \\big)\n",
    "$$\n",
    "\n",
    "[<div align=\"center\"><img src=\"figures/2/delta_spp.PNG\" width=\"300\"/></div>](delta_spp.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Noise in sensors\n",
    "\n",
    "So far, we have been talking about noise in a measurement.  In sensing, \n",
    "typically one measures the \"thing\" we want to sense by translating it into\n",
    "a signal that one can record, like a voltage or a current.  The device that\n",
    "converts the quantity that we want to measure, such as the position of an object\n",
    "or the power of a laser, into an observable signal is called a \"sensor\":\n",
    "\n",
    "[<div align=\"center\"><img src=\"figures/2/sensor.PNG\" width=\"500\"/></div>](sensor.png)\n",
    "\n",
    "A sensor often produces an output that is linearly proportional to the\n",
    "input. For example, if the sensors input is position $x$, and the output of the sensor is a voltage, then we would have: \n",
    "\n",
    "$$\n",
    "V = R x \\quad R = \\frac{dV}{dx}\n",
    "$$\n",
    "\n",
    "Here, we have defined a constant $R$ as the conversion factor, sometimes called\n",
    "\"transduction gain\", or also the \"responsivity\". In this example it could have units such as Volts / mm, describing how many volts the sensor outputs for a millimeter displacement of th object is measuring. \n",
    "\n",
    "More generally, the reponsivity $R$ is the proportionality constant (or slope) relating the signal out $S_out$ of the sensor to the signal coming into it $S_in$:\n",
    "\n",
    "$$\n",
    "R = \\frac{dS_{out}}{dS_{in}}\n",
    "$$\n",
    "\n",
    "Another example of a sensor could be, for example, a voltage amplifier which takes a voltage $V_{in}$ at its input and amplifies this to create a (larger) voltage at its output $V_{out}$:\n",
    "\n",
    "$$\n",
    "R = \\frac{dV_{out}}{dV_{in}} \n",
    "$$\n",
    "\n",
    "Note that in this example, since the output and the input have the same units, the responsivity is in fact the voltage gain $G$ of the amplifier. \n",
    "\n",
    "How sensitive is our sensor?  A common mistake is to associate $R$ with the\n",
    "sensitivity, as in: \"A photodetector with a larger Amps/Watt conversion factor is\n",
    "more sensitive\".\n",
    "\n",
    "If we define \"sensitivity\" as the ability to distinguish a small signal from \n",
    "the noise, then we can see that this is nonsense!\n",
    "\n",
    "Why?  Because our ability to distinguish signals from noise *also* depends on\n",
    "how much noise we have in the reading of our sensor: what we care about is the\n",
    "equivalent $\\sigma_x$ given the amount of noise $\\sigma_v$ we have in our\n",
    "recorded data. For the linear sensor described above, these are related as\n",
    "\n",
    "$$\n",
    "\\sigma_x = \\frac{\\sigma_v}{R_{vx}} = \\frac{\\sigma_v}{dV/dx}\n",
    "$$\n",
    "\n",
    "```{admonition} Sensitivity\n",
    ":class: note\n",
    "The sensitivity of our detector is determined by the power spectral density\n",
    "of its noise\n",
    "\n",
    "$$\n",
    "S_{xx} = \\frac{S_{vv}}{(R_{vx})^2}\n",
    "$$\n",
    "\n",
    "Typically, when discussing sensors, people prefer to work with \"sensitivity\"\n",
    "defined as:\n",
    "\n",
    "$$\n",
    "\\sqrt{S_{xx}} = \\frac{\\sqrt{S_{vv}}}{R_{vx}}\n",
    "$$\n",
    "\n",
    "For a sensor of position, the sensitivity $\\sqrt{S_{xx}}$ has units of \n",
    "m/$\\sqrt{Hz}$.\n",
    "```\n",
    "\n",
    "What does $\\sqrt{S_{xx}}$ tell us?  If $\\sqrt{S_{xx}}$ =\n",
    "1nm/$\\sqrt{Hz}$, then it means if we average for 1 second, we will be able to\n",
    "detect a 1nm displacement signal with a signal to noise (SNR) ratio of 1:\n",
    " \n",
    "[<div align=\"center\"><img src=\"figures/2/sensor_noise.PNG\" width=\"500\"/></div>](sensor_noise.png)\n",
    "\n",
    "> One slightly confusing point: better sensitivity is *smaller* $\\sqrt{S_{xx}}$.\n",
    "\n",
    "One last point: what if the relation between $x$ and $V$ is not linear? If the \n",
    "noise is small compared to the signal, then we can use the rules of error\n",
    "propagation to find the conversion of $\\sigma_v$ into $\\sigma_x$:\n",
    "\n",
    "$$\n",
    "V = f(x)\\\\\n",
    "\\sigma_v = \\frac{df}{dx} \\sigma_x\n",
    "$$\n",
    "\n",
    "It looks similar, but now the derivative may be dependent on the average value\n",
    "of $x$ (i.e. not just constant).\n",
    "\n",
    "```{admonition} Conclusions\n",
    ":class: important\n",
    "\n",
    "1. The central limit theorem tells us that the probability distribution of an infinite sum of independent random variables is Gaussian  \n",
    "2. A Poissonian probability distribution arises when events happen independently\n",
    "3. The sensitivity of a detector is limited by the power spectral density of the detector noise \n",
    "```"
   ]
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