- •1. TABLE OF CONTENTS
- •2. MATHEMATICAL TOOLS
- •2.1 INTRODUCTION
- •2.1.1 Constants and Other Stuff
- •2.1.2 Basic Operations
- •2.1.2.1 - Factorial
- •2.1.3 Exponents and Logarithms
- •2.1.4 Polynomial Expansions
- •2.2 FUNCTIONS
- •2.2.1 Discrete and Continuous Probability Distributions
- •2.2.2 Basic Polynomials
- •2.2.3 Partial Fractions
- •2.2.4 Summation and Series
- •2.3 SPATIAL RELATIONSHIPS
- •2.3.1 Trigonometry
- •2.3.2 Hyperbolic Functions
- •2.3.2.1 - Practice Problems
- •2.3.3 Geometry
- •2.3.4 Planes, Lines, etc.
- •2.4 COORDINATE SYSTEMS
- •2.4.1 Complex Numbers
- •2.4.2 Cylindrical Coordinates
- •2.4.3 Spherical Coordinates
- •2.5 MATRICES AND VECTORS
- •2.5.1 Vectors
- •2.5.2 Dot (Scalar) Product
- •2.5.3 Cross Product
- •2.5.4 Triple Product
- •2.5.5 Matrices
- •2.5.6 Solving Linear Equations with Matrices
- •2.5.7 Practice Problems
- •2.6 CALCULUS
- •2.6.1 Single Variable Functions
- •2.6.1.1 - Differentiation
- •2.6.1.2 - Integration
- •2.6.2 Vector Calculus
- •2.6.3 Differential Equations
- •2.6.3.1 - First Order Differential Equations
- •2.6.3.1.1 - Guessing
- •2.6.3.1.2 - Separable Equations
- •2.6.3.1.3 - Homogeneous Equations and Substitution
- •2.6.3.2 - Second Order Differential Equations
- •2.6.3.2.1 - Linear Homogeneous
- •2.6.3.2.2 - Nonhomogeneous Linear Equations
- •2.6.3.3 - Higher Order Differential Equations
- •2.6.3.4 - Partial Differential Equations
- •2.6.4 Other Calculus Stuff
- •2.7 NUMERICAL METHODS
- •2.7.1 Approximation of Integrals and Derivatives from Sampled Data
- •2.7.2 Euler First Order Integration
- •2.7.3 Taylor Series Integration
- •2.7.4 Runge-Kutta Integration
- •2.7.5 Newton-Raphson to Find Roots
- •2.8 LAPLACE TRANSFORMS
- •2.8.1 Laplace Transform Tables
- •2.9 z-TRANSFORMS
- •2.10 FOURIER SERIES
- •2.11 TOPICS NOT COVERED (YET)
- •2.12 REFERENCES/BIBLIOGRAPHY
- •3. WRITING REPORTS
- •3.1 WHY WRITE REPORTS?
- •3.2 THE TECHNICAL DEPTH OF THE REPORT
- •3.3 TYPES OF REPORTS
- •3.3.1 Laboratory
- •3.3.1.1 - An Example First Draft of a Report
- •3.3.1.2 - An Example Final Draft of a Report
- •3.3.2 Research
- •3.3.3 Project
- •3.3.4 Executive
- •3.3.5 Consulting
- •3.3.6 Interim
- •3.4 ELEMENTS
- •3.4.1 Figures
- •3.4.2 Tables
- •3.4.3 Equations
- •3.4.4 Experimental Data
- •3.4.5 References
- •3.4.6 Acknowledgments
- •3.4.7 Appendices
- •3.5 GENERAL FORMATTING
- •Title: High Tech Presentations The Easy Way
- •1.0 PRESENTATIONS IN GENERAL
- •2.0 GOOD PRESENTATION TECHNIQUES
- •2.1 VISUALS
- •2.2 SPEAKING TIPS
- •3.0 PRESENTATION TECHNOLOGY
- •3.1 COMMON HARDWARE/SOFTWARE
- •3.2 PRESENTING WITH TECHNOLOGY
- •X.0 EXAMPLES OF PRESENTATIONS
- •4.0 OTHER TECHNOLOGY ISSUES
- •4.1 NETWORKS
- •4.1.1 Computer Addresses
- •4.1.2 NETWORK TYPES
- •4.1.2.1 Permanent Wires
- •4.1.2.2 Phone Lines
- •4.1.3 NETWORK PROTOCOLS
- •4.1.3.1 FTP - File Transfer Protocol
- •4.1.3.2 HTTP - Hypertext Transfer Protocol
- •4.1.3.3 Novell
- •4.1.4 DATA FORMATS
- •4.1.4.1 HTML - Hyper Text Markup Language
- •4.1.4.1.1 Publishing Web Pages
- •4.1.4.2 URLs
- •4.1.4.3 Hints
- •4.1.4.4 Specialized Editors
- •4.1.4.6 Compression
- •4.1.4.7 Java
- •4.1.4.8 Javascript
- •4.1.4.9 ActiveX
- •4.1.4.10 Graphics
- •4.1.4.11 Animation
- •4.1.4.12 Video
- •4.1.4.13 Sounds
- •4.1.4.14 Other Program Files
- •4.2 PULLING ALL THE PROTOCOLS AND FORMATS TOGETHER WITH BROWSWERS
- •REFERENCES
- •AA:1. ENGINEERING JOKES
- •AA:1.1 AN ENGINEER, A LAWYER AND A.....
- •AA:1.2 GEEKY REFERENCES
- •AA:1.3 QUIPS
- •AA:1.4 ACADEMIA
- •AA:1.4.1 Other Disciplines
- •AA:1.4.2 Faculty
- •AA:1.4.3 Students
- •AA:1.5 COMPUTERS
- •AA:1.5.1 Bill
- •AA:1.5.2 Internet
- •AA:1.6 OTHER STUFF
- •2. PUZZLES
- •2.1 MATH
- •2.2 STRATEGY
- •2.3 GEOMETRY
- •2.4 PLANNING/DESIGN
- •2.5 REFERENCES
- •3. ATOMIC MATERIAL DATA
- •4. MECHANICAL MATERIAL PROPERTIES
- •4.1 FORMULA SHEET
- •5. UNITS AND CONVERSIONS
- •5.1 HOW TO USE UNITS
- •5.2 HOW TO USE SI UNITS
- •5.3 THE TABLE
- •5.4 ASCII, HEX, BINARY CONVERSION
- •5.5 G-CODES
- •6. COMBINED GLOSSARY OF TERMS
page 81
2.9 z-TRANSFORMS
∞
• For a discrete-time signal x[n ], the two-sided z-transform is defined by .X(z ) = ∑ x[n ]z–n
n = –∞
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The one-sided z-transform is defined by .XIn(zboth=) |
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cases,x[then z]z-transform is |
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• The inverse z-transform is obtained by contour integration in the complex plane |
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j2π |
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similar to the Laplace transform.
• Along with a z-transform we associate its region of convergence (or ROC). These are the values of z for which Xis(zbounded) (i.e., of finite magnitude).
page 82
• Some common z-transforms are shown below.
Table 1: Common z-transforms
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z-Transform |
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ROC |
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X(z ) |
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All z |
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u[n ] |
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z |
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>1 |
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1 – z–1 |
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nu[n ] |
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z–1 |
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(1 – z |
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n2u[n ] |
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z–1(1 + z |
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(1 – z |
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u[n ] |
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1 – az–1 |
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nanu[n ] |
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az–1 |
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(1 – az |
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(–a |
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(–nan )u[– n – 1 |
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cos (ω 0n )u[n ] |
1 – z–1 cos ω |
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1------------------------------------------------– 2z–1 cos ω |
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sin (ω 0n )u[n ] |
z–1 sin ω |
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1------------------------------------------------– 2z–1 cos ω |
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cos (ω 0n )u[n |
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1 – az–1 cos ω 0 |
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1 – 2az–1 cos ω |
0 + a2z–2 |
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page 83
Table 1: Common z-transforms
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Signal |
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z-Transform |
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ROC |
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x[n ] |
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X(z ) |
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sin (ω 0n )u[n |
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az–1 sin ω 0 |
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1 – 2az–1 cos ω |
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k!(n – k )!u[n ]---------------------- |
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• The z-transform also has various properties that are useful. The table below lists properties for the two-sided z-transform. The one-sided z-transform properties can be derived from the ones below by considering the signal x[n ]u[n ]instead of simply x[n ].
Table 2: Two-sided z-Transform Properties
Property |
Time Domain |
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Notation |
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X(z ) |
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α x1[n ]+ xβ2[n ] |
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Differentiation |
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–z------------- |
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page 84
Table 2: Two-sided z-Transform Properties
Property |
Time Domain |
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Convolution |
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Multiplication |
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------- X1 |
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j2π ° |
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lim X(z ) |
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rem |
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2.10 FOURIER SERIES
•These series describe functions by their frequency spectrum content. For example a square wave can be approximated with a sum of a series of sine waves with varying magnitudes.
•The basic definition of the Fourier series is given below.
a0 |
∞ |
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f(x ) = ---- + |
2
n = 1
an
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an cos nπ |
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+ bn sin |
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1 L |
f(x )cos |
nπ x |
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nπ x |
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dx |
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f(x )sin |
-------- |
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2.11 TOPICS NOT COVERED (YET)
• To ensure that the omissions are obvious, I provide a list of topics not covered below. Some of these may be added later if their need becomes obvious.
• Frequency domain - Fourier, Bessel