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 threehour 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; SturmLiouville 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 powerlaw distributions
 Hypothesis testing; idea of test statistics; chisquared statistic; Fstatistic
 Parameter estimation; properties of estimators; maximum likelihood methods; weighted mean and variance; minimum chisquared 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 chisquared or maximum credibility; multiparameter confidence intervals; interesting and uninteresting parameters

Information for Visiting Students
Prerequisites  None 
High Demand Course? 
Yes 
Course Delivery Information

Academic year 2023/24, 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)   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 nonmathematical 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.

Additional Information
Graduate Attributes and Skills 
Not entered 
Keywords  FASt 
Contacts
Course organiser  Prof Graeme Ackland
Tel: (0131 6)50 5299
Email: gjackland@ed.ac.uk 
Course secretary  Ms Lucy DavisJenkins
Tel:
Email: ldavisj@ed.ac.uk 

