"Calculus-level (Programming) Language and Apps" 3 posts Sort by created date Sort by defined ordering View as a grid View as a list

Optimum Matched Filter Design (Transfer Function)

        Optimum Matched Filter (Transfer Function)

(Nested Processes ... Each Process controlled by a Solver)

Problem Description

The transfer function H(s) is the Laplace transform of the output signal Yout(s)* divided by the Laplace transform of the input signal Yin(s)*: that is H(s) = Yout(s) / Yin(s)  where each signal's transform is assumed to be a ratio of polynomials. 

For more, read attached PDF file.

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Engineering Design Optimization using Calculus Level Methods: A Casebook Approach (Manual)

 2011 Optimal Designs Enterprise 1
Engineering Design Optimization using Calculus Level Methods: A Casebook Approach
By Phil B Brubaker

                    Welcome to Calculus-level Problem Solving!

Engineers in industry wanted to ‘tweak’ their parameters. So this textbook was written to show the simplicity of ‘tweaking’ parameters in algebraic through differential equation problems when using a Calculus-level language like PROSE or FortranCalculus. FortranCalculus (FC) is available on the web.

Automatic Differentiation (AD) and Operator overloading were key technologies that allowed numerical methods, now called solvers, to be stored in a FC library. A user will use a solver by stating a solver name in a ‘find’ statement using the ‘by’ clause. Want to switch solvers? Just change the solver name (e.g. from ‘Ajax’ to ‘Jupiter’) and you are ready to try a different numerical method! It is that easy to code. (See the FortranCalculus manual for suggestions on what solver to use for a given problem.)

Help spread the word about Calculus-level thinking and problem solving. Do you know any
engineering or science professors that might have a problem that could be solved and shown to their future students?

This textbook tries to move today’s thinking from solving one problem at a time, to solving all of their project’s problems at once while tweaking parameters in order to achieve an optimum solution. This requires Calculus-level thinking. An analogy might be thinking in terms of Machine code, one bit at a time. Today, computer simulations have people thinking in terms of Algebraic code, one problem at a time. We are trying to move people to Calculus-level code, solving entire projects at a time. This will reduce development time and improve accuracy of their math models. (Future CEOs should study the Oil Refinery Production problem in order to see future possibilities with Calculus-level thinking.)

Mission Statement:
Get the FortranCalculus compiler operational and in use via the internet. It’s a free compiler that simplifies solving math problems by minimizing code necessary to state & solve a problem. Some new thinking is necessary for those wanting to get the most for their buck; convert from simulation to optimization thinking.

What’s the different between simulation and optimization? Picture a saw horse construction project. A Simulation would yield A saw horse where Optimization would yield an Optimal saw horse. If the objective (function) was good and proper then the Optimal saw horse would be the best solution, right? For example, the objective might be lightest & strongest saw horse. A wrong objective might be just the strongest saw horse. This might yield a strong horse but a very heavy one!

If you were a manager or CEO and had the choice of a simulation design versus an optimization design, which would you pick?

Modeling & Simulation’s next step is (Mathematical) Optimizations. Optimizations require an Objective (function). Today's Engineers & Scientists solve problems with a “Find X” mind-set. With some Operational Research training they could expand their thinking to a “Find X to Optimize Y” mind-set. Then they would be ready for Optimizations, Calculus-level programming and software. (This would drop today’s design times that require months even man years to one or two days! Manufacturing processes could be optimized to the days demand and thus maximize their profits.)

“Find X to Optimize Y” thinking among professors will cause most Engineering & Science textbooks to be rewritten with optimization examples and discussions. This will be great stuff for industries and government; applied engineering and/or science not just theories.

Table of Content:
Welcome to Calculus-level Problem Solving! .......................................................................................1
About .....................................................................................................................................5
Introduction........................................................................................................................................5
1 General Algebraic Equations .............................................................................................................9
Background of TFH Math Model for a Readback Pulse from Magnetic Recording...................9
A Typical Readback Pulse from Magnetic Recording ..............................................................12
An Unusual Readback Pulse from Magnetic Recording...........................................................15
A Typical Readback Pulse from Magnetic Recording with Improved Model ............................17
An Unusual Readback Pulse from Magnetic Recording with Improved Model.........................19
Curve fitting: A Sinusoidal Signal...........................................................................................21
Curve fitting: A Damped Sinusoidal Signal .............................................................................23
1.4 Conclusion on Curve Fitting..............................................................................................25
Pharmacokinetics....................................................................................................................26
Slack Variable Techniques......................................................................................................29
Paper Bicycle Design..............................................................................................................31
Chapter 1 Exercises ................................................................................................................33
2 La Place Transforms..........................................................................................................................34
Optimum Matched Filter (Transfer Function) ..........................................................................34
Chapter 2 Exercises ................................................................................................................44
3 Ordinary Differential Equations .........................................................................................................46
Second Order Non-Linear ODE ..............................................................................................47
A Third Order Non-Linear ODE .............................................................................................50
A Bang-Bang Control Problem ...............................................................................................53
Non-Linear Equations of Motion.............................................................................................62
4 System of Differential Equations .......................................................................................................65
The Lorentz Equations, a System of ODEs..............................................................................66
The Convection Reaction Equations, a System of PDEs ..........................................................69
Body Plasma Chemistry..........................................................................................................71
Modeling a Nanostructured Solar Cell.....................................................................................76
Chapter 4 Exercises ................................................................................................................82
5 Partial Differential Equations.............................................................................................................83
PDEs: Stock Market to Biology ..............................................................................................84
Burgers’ Equation...................................................................................................................86
Telegrapher’s Equation ...........................................................................................................89
6 Inverse Problems ...............................................................................................................................92
Custom Thermistor Design .....................................................................................................93
Drug Development..................................................................................................................95
Heat Transfer over 1D Slab.....................................................................................................96
Robot Arm Movement ............................................................................................................99
Plane Crash Locator................................................................................................................102
7 Implicit Equations .............................................................................................................................104
System of Implicit Algebraic Equations ..................................................................................105
2nd Order Implicit Differential Equation ..................................................................................108
8 Nesting Solvers .................................................................................................................................110
Nesting … Matched Filter.......................................................................................................111
Oil Refinery Production ..........................................................................................................113
9 Miscellaneous....................................................................................................................................118
Monte Carlo Simulation OR Total Derivative? ........................................................................118
Stiff Equations & Trouble Shooting ........................................................................................119
10 Conclusions.....................................................................................................................................121
10.1 Future: Thinking outside the box .....................................................................................121
11 Appendix.........................................................................................................................................125

Picking the right Solver...........................................................................................................125
‘aplot’ source code..................................................................................................................125
Spectral Estimation (freeware) Software..................................................................................126
‘readrit1.100’ File Listing .......................................................................................................126
‘readrit2.200’ File Listing .......................................................................................................127
Arbitrary Equalization with Simple LC Structures ...................................................................130
Incomplete Problems: can you help complete one or more? .....................................................133
Index....................................................................................................................................................134

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For more, read the attached PDF file
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PS: Have you seen movie Hidden Figures, if you haven’t seen it, you should.  It may help understand the communication problem between different types of people; e.g. NASA’s Engineers, Scientists, and Mathematicians.  The movie shows engineers trying to solve some equations and getting no where fast!  A mathematician comes along and solves the problem.  But numbers say little to engineers.  Near the movie’s end, the mathematician draws a graph showing a solution.  The engineers finally get the ‘picture’ of what the equations are trying to say to them. Remember, a picture is worth a 1,000 words, right?

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Inverse Problems solved with a Calculus-level (Programming) Language

                                                    Inverse Problems

“You know what you want, just don’t know how to get there.”

Inverse Problems (IPs) are an important special set of problems that fit the statement “You know what you want, you just don’t know how to get there.”  A proper directive can get an Inverse Problem solved in hours!  Here are some examples of Inverse Problems:

  1. Law enforcement: Shells found at scene where did it come from?
  2. Airplane crash with wreckage all over the place.  How did these parts get where they lay?
  3. Missile target: Have target, how to get missile there?
  4. Want a ‘black box’ to have an efficiency of 54.321%.  How to design/build such a black box?
  5. Car seat storage H x W x D slot, how to design seat so as it will fit into slot while maximizing seat comfort?

A Parameter Estimation for an Inverse Problem is solved using the Calculus-level Find statement shown here:

Find a   ooo   To Match Error

If a Bound Value Problem then: Find a, ydot0, y2dot0   ooo   To Match Error

Where ‘a’ may be a vector with ‘n’ parts, a1, a2, a3,…an;
ydot0, y2dot0,etc. are derivatives at independent variable = 0; and,
‘error’ is the objective function.

If the IP problem contains any Ordinary or Partial Differential Equations (ODEs or PDEs) , the ‘find’ statement is wrapped around an integrate & integration statement in order to solve the ODEs or PDEs while finding the best ‘a’ parameter(s) for the given problem.

The ‘a’ parameter(s) are varied to fit one’s ‘m’ data points that make up the objective function, error.  This technique can vary as many parameters as you want; e.g. 5 or 50 or 50,000.  If there are less equations than parameters m < n, this would be classified as an under-determined system of equations.  If there are more equations than parameters, m > n, this would be an over-determined system.  Under- or Over-determined systems might force one to switch solvers to do the job.

--- for more, read attachment file.

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