Undergraduate Course: Fourier Analysis and Statistics (PHYS09055)

Course Outline
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
SCQF Credits	20	ECTS Credits	10
Summary	A coherent 20pt course taken by all single honours physics students. Examined via a single three-hour paper in the December diet.
Course description	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

Entry Requirements (not applicable to Visiting Students)
Pre-requisites	Students MUST have passed: ( Linear Algebra and Several Variable Calculus (PHYS08042) AND Dynamics and Vector Calculus (PHYS08043) AND Practical Physics (PHYS08048)) AND Algebra and Calculus (PHYS08041)	Co-requisites
Prohibited Combinations	Students MUST NOT also be taking Fourier Analysis (PHYS09054)	Other requirements	None
Additional Costs	None

Information for Visiting Students
Pre-requisites	None
High Demand Course?	Yes

Course Delivery Information

Academic year 2015/16, Available to all students (SV1)		Quota: None
Course Start	Semester 1
Timetable	Timetable
Learning and Teaching activities (Further Info)	Total Hours: 200 ( Lecture Hours 22, Seminar/Tutorial Hours 22, Formative Assessment Hours 3, Programme Level Learning and Teaching Hours 4, Directed Learning and Independent Learning Hours 149 )
Assessment (Further Info)	Written Exam 80 %, Coursework 20 %, Practical Exam 0 %
Additional Information (Assessment)	Coursework 20% and examination 80%.
Feedback	Not entered
Exam Information
Exam Diet	Paper Name		Hours & Minutes
Main Exam Diet S1 (December)	Fourier Analysis and Statistics		3:00

Academic year 2015/16, Part-year visiting students only (VV1)		Quota: None
Course Start	Semester 1
Timetable	Timetable
Learning and Teaching activities (Further Info)	Total Hours: 200 ( Lecture Hours 22, Seminar/Tutorial Hours 22, Formative Assessment Hours 3, Programme Level Learning and Teaching Hours 4, Directed Learning and Independent Learning Hours 149 )
Assessment (Further Info)	Written Exam 80 %, Coursework 0 %, Practical Exam 20 %
Additional Information (Assessment)	Coursework 20% and examination 80%.
Feedback	Not entered
Exam Information
Exam Diet	Paper Name		Hours & Minutes
Main Exam Diet S1 (December)	Fourier Analysis and Statistics		3:00

Learning Outcomes
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.