" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "## Part 1: Finding your location\n", "\n", "### How does a GPS chip determine where it is located?\n", "\n", "Let's explore what a GPS chip does: we'll build up a simple model, and then let you pretend to be the GPS chip.\n", "\n", "\n", "\n", "We first assume that we have obtained the $N$ GPS signals, each of which gives a noisy measurement of the distance between the GPS satellite and the object.\n", "\n", "The noisy measurements are modeled as follows, where $n_i$ is iid Gaussian noise with zero mean and variance $\\sigma^2$. $$D_i = d_i + n_i $$\n", "\n", "In the above equation, $d_i$ is the actual distance to the $i$-th satellite, and $D_i$ is the reported distance, which is corrupted by additive noise $n_i$. This additive white gaussian noise (AWGN) channel model is actually very common in information theory, and can be analyzed just like how we analyzed the BEC and BSC earlier in the course! It is a pretty good model for satellite communication links as you don't have to deal with shadowing, multipath, excessive interference, etc. (come talk to one of us if you're interested in learning more about this stuff/what it means!)\n", "\n", "For simplicity, let's visualize the entire space as a 2D plane. Assume that all GPS satellites and the object to be located (GPS chip) are on the plane. Denote the position of the unknown object as $(x, y)$, and let $(x_i, y_i)$ be the position of the $i$th GPS satellite.\n" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "### Q1. Find the MLE of $(x, y)$ given $\\{(x_i,y_i)\\}$ and $\\{D_i\\}$.\n", "\n", "Hint 1: To get started thinking about the problem, consider the case where $\\sigma=0$, i.e., noiseless distance measures are given. What is the minimum $N$ necessary to estimate the unknown position exactly, and how would you estimate it?\n", "\n", "Hint 2: Leave your answer in the form of an expression to be maximized (as in, something proportional to the likelihood function)\n", "\n", "Hint 3: Use the distance formula. What is the relationship betweek $d_i$ and $(x_i,y_i)$?" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "#### Q1. Answer here" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "

" ] }, { "cell_type": "markdown", "metadata": {}, "source": [ "\n", "\n", "### Subject: [cory-info] CORY HALL BURGLARY YESTERDAY NIGHT BETWEEN 11:03 P.M.- 11:44 P.M.\n", "### Date: Wed, April 6, 2016 at 10:19 AM\n", "\n", "Dear Building Occupants:\n", "\n", "Yesterday night between 11:03 p.m. and 11:44 p.m. the BLISS Lab was burglarized. The elapsed time for entry, theft and exit from the building was approximately 6 minutes. To prevent thefts from occurring inside or nearby Cory Hall and Soda Hall please remember to:\n", "- Be AWARE of your surroundings; Be aware;\n", "- Lock up all personal belongings when you leave the building\n", "- Never prop doors open allowing individuals without card key access to enter a secure space\n", "\n", "Don’t allow strangers to “tail gate” behind you through card reader controlled doors. Immediately report any suspicious activity to UCPD at (510) 642-6760 and myself at (415) 713-3403.\n", "\n", "##### BE SAFE, REMAIN VIGILIGENT and AWARE.\n", "\n", "======================================================================================================================\n", "\n", "Indeed, the hidden secret of EE126 is stolen from Kangwook’s desk in the BLISS lab. The note has been secretly shared among teaching staffs at Berkeley for more than 50 years, and has been secret sauce of EE126. \n", "\n", "