Audio Compression: MP3
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1 Audio Compression: MP3 Problem: Digital Music Recording huge data amounts! Cassandra Wilson: Come On In My Kitchen (MP3, 4.4 MB) Cassandra Wilson: Come On In My Kitchen (WAF, 49.1 MB) Compression Factor: 11.2 Audio Compression: MP3 Sampling: How CDs work Tones are Waves 1
2 Audio Compression: MP3 Compression: The NA part of MP3 Expand the sampled function into a series (FFT): Simplification: Discard some of the series elements (coefficients) Use Perceptual Model Huffman encoding of the coefficients Etc. GPS Navigation Problem: How to know, where you are? Easy But: How Solution: does the Ask GPS your unit GPS unit! know? Can I trust the result? 2
3 GPS Navigation GPS Space Segment: 27 (or so) Satellites orbiting Earth GPS Control Segment: Based in Colorado P. H. Dana, GPS Navigation GPS User Segment: P. H. Dana, 3
4 GPS Navigation GPS User Segment: Solve system of linear equations/least squares problem P. H. Dana, Numerical Weather Prediction Problem: Given initial data, what is the weather tomorrow? Satellite Forecast Picture 10/09/ /08/ /07/
5 Numerical Weather Prediction Mathematical Model: Set of PDEs Primitive Equations Momentum eq. Hydrostatic approx. Continuity eq. Tracer eq. for Moisture/Temp. Eq. of state Numerical Weather Prediction Simplification: Barotropic Equations Momentum eq. Continuity eq. Discretization: Differential eq. difference eq. 5
6 Numerical Weather Prediction Transformation: Towards System of Equations Transformation: Stationary Problem Discretization: Sparse (large) system of equations Numerical Weather Prediction Discretization: Grids 6
7 Numerical Weather Prediction Simulation: Simplified simulation: flow over mountain M. Läuter (2003) Summary Three Examples: Methods from NA used Fast Fourier Transform (Discrete Series Expansion) Least Squares Approximation and Error Correction Finite Difference Approximation of Differential Operators (Numerical Solution of Partial Differential Equations) Gridding Techniques Solution of Large Systems of Equations MP3 Example GPS Example NWP Example Subject of 2 nd Term 7
8 Failure: Patriot Missile What Happened? During first Gulf War: American Patriot battery failed 28 Soldiers died 100 other persons injured How Patriot works 1. Detect object (e.g. Scud) Calculate range gate 2. Confirmation 3. Tracking Failure: Patriot Missile Analysis Range gate calculation from velocity and time Time in 1/10 seconds (integer) 24 bit registers in Patriot computer Time conversion from integer to floating point Velocity is floating point 8
9 Failure: Sleipner Oil Rig Sleipner A Platform Oil/Gas producing platform in North Sea Support on sea bed 82m depth Concrete Base of 16,000 m 2, 24 cells First Concrete Base sprang a leak and sank 08/23/1991 (700 Mio. USD loss) Failure: Sleipner Oil Rig What Happened? Conrete Base made of 24 cells Cells connected by tricells Tricell sprang One tricell failure caused a chain reaction Concrete cell structure Broken tricell Sketch of concrete base Seminar-Softwarefehler/Folien/schaden.pdf 9
10 Failure: Sleipner Oil Rig Standard FEM Software NASTRAN Underestimates forces by 47% Concrete cell structure FEM Mesh for tricell FEM Mesh for concrete base Summary Two Examples: Incorrect usage of NA leads to disaster Round-off error Floating point representation of numbers Insufficient resolution Insufficient accuracy checking Patriot Example Sleipner Example 10
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