Undergraduate Course: Fourier Analysis and Statistics (PHYS09055)
|School||School of Physics and Astronomy
||College||College of Science and Engineering
|Credit level (Normal year taken)||SCQF Level 9 (Year 3 Undergraduate)
||Availability||Available to all students
|Summary||A coherent 20pt course taken by all single honours physics students. Examined via a single three-hour paper in the December diet.
Fourier Analysis (20 lectures)
- Fourier series: sin and cos as a basis set; calculating coefficients; complex basis; convergence, Gibbs phenomenon
- Fourier transform: limiting process; uncertainty principle; application to Fraunhofer diffraction
- Dirac delta function: Sifting property; Fourier representation
- Convolution; Correlations; Parseval's theorem; power spectrum
- Sampling; Nyquist theorem; data compression
- Solving Ordinary Differential Equations with Fourier methods; driven damped oscillators
- Green's functions for 2nd order ODEs; comparison with Fourier methods
- Partial Differential Equations: wave equation; diffusion equation; Fourier solution
- Partial Differential Equations: solution by separation of variables
- PDEs and curvilinear coordinates; Bessel functions; Sturm-Liouville theory: complete basis set of functions
Probability and Statistics (20 lectures)
- Concept and origin of randomness; randomness as frequency and as degree of belief
- Discrete and continuous probabilities; combining probabilities; Bayes theorem
- Probability distributions and how they are characterised; moments and expectations; error analysis
- Permutations, combinations, and partitions; Binomial distribution; Poisson distribution
- The Normal or Gaussian distribution and its physical origin; convolution of probability distributions; Gaussian as a limiting form
- Shot noise and waiting time distributions; resonance and the Lorentzian; growth and competition and power-law distributions
- Hypothesis testing; idea of test statistics; chi-squared statistic; F-statistic
- Parameter estimation; properties of estimators; maximum likelihood methods; weighted mean and variance; minimum chi-squared method; confidence intervals
- Bayesian inference; priors and posteriors; maximum credibility method; credibility intervals
- Correlation and covariance; tests of correlation; rank correlation test; least squares line fitting
- Model fitting; analytic curve fitting; numerical model fitting; methods for finding minimum chi-squared or maximum credibility; multi-parameter confidence intervals; interesting and uninteresting parameters
Information for Visiting Students
|High Demand Course?
Course Delivery Information
|Academic year 2022/23, Available to all students (SV1)
|Learning and Teaching activities (Further Info)
Lecture Hours 22,
Seminar/Tutorial Hours 22,
Formative Assessment Hours 3,
Programme Level Learning and Teaching Hours 4,
Directed Learning and Independent Learning Hours
|Assessment (Further Info)
|Additional Information (Assessment)
||Coursework 20% and examination 80%.
||Hours & Minutes
|Main Exam Diet S1 (December)||3:00|
On completion of this course, the student will be able to:
- State in precise terms key concepts relating to Fourier analysis and probability & statistics.
- Master the derivations of a set of important results in Fourier analysis and probability & statistics.
- Apply standard methods of Fourier analysis and probability & statistics to solve unseen problems of moderate complexity.
- Understand how to take a physical problem stated in non-mathematical terms and express it in a way suitable for applying the tools of this course.
- Be able to think critically about the results of solving such problems: whether they make sense physically, and whether different mathematical approaches are equivalent.
|Graduate Attributes and Skills
|Course organiser||Prof Andy Lawrence
Tel: (0131) 668 8346
|Course secretary||Miss Hayley Crawford
Tel: (0131 6)51 7524