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Course Wrapup Physics 3730/6720 Spring Semester 2016 Final Project Comments Critical Frequency Critical sampling (Nyquist) of sine wave is two per cycle. fc =1/(2t) Frequencies above this are undersampled; discrete FT is in trouble! Figures from Stephen Smith, Digital Signal Processing Discrete Fourier Transforms ● ● Suppose that we have N sampled values (further, suppose N is even)... Now note that with N numbers input, we can produce at most N numbers output. (Why?) The most interesting range is -fc→ fc, so we'll want to estimate frequency components at the discrete values: Intensity #bins = #bins in time domain -0.00833 0.0 f(Hz) +0.00833 Announcements ● ● Due to 3740 final, Final Project due date is pushed back to Sunday May 1st Midnight th I will be in SP 205 at 9:00 AM April 28 for consultation about project, final exam, etc. Final Exam Wednesday May 4 , 8:00-10:00 AM ● Similar format to midterm: ● th ● – Open book, notes, internet – Consult only with instructor – Online submission a la assignments. nd Focus on 2 half of semester, but “comprehensive” as needs be. Midterm Topics ● (Basic Unix and Emacs) ● sort, awk, sed ● Plotting with gnuplot ● Basic shell scripts ● Basic html ● C++ ● – math operations – input/output – conditionals, loops – pointers, references – random number generators Program and plot with Python Potential (Major) Final Exam Topics ● Statistics; Binomial, Poisson, Gaussian ● Numerical integration; trapezoid, Monte Carlo... ● Least-squares fitting ● Likelihood fitting ● ODEs (Euler, Runge-Kutta...) ● Discrete Fourier Transforms ● Maple (plotting, Matrix algebra) ● LaTex ● Python Binomial Distribution ● ● ● The probability PB of observing successes in N trials, where the probability of success per trial is p, is given by: The average number of successes: The standard deviation of the number of successes is given by: Midpoint Rule ● ● ● Divide x-axis into intervals of width h. Use midpoint of each interval to compute area. Error The “Monte Carlo” Technique ● ● ● Employ's random numbers to determine integrals. e.g. what is the area of a circle? What is ? = 4x(709/900) = 3.151 ± 0.055 Evaluating Fits: ● ● ● 2 Definition of 2 Absolute magnitude of 2 only has meaning relative to the number of degrees of freedom #DOF = (# of data points) (# parameters determined from the data) ● #DOF = (# of data points) - (# constraints) ● 2 /#DOF ~ 1 for good fit LaTex Math I can put mathematical characters in a sentence by putting a dollar sign before and after the characters. The area $A$ of a circle is given by $A = \pi r^2$. I can also set equations apart from the text as in the following example: \begin{equation} A = \pi r^2 \end{equation} Note that latex will keep track of equation numbers for me. Integrate ODEs: Graphical Explanation ● ● ● A is initial point. Euler uses derivative at A to find approximation to y(t+h) at B Modified Euler uses average of derivatives at A and B to find better approximation C