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# Postgraduate Course: Statistical Signal Processing (SSP) (MSc) (PGEE11122)

 School School of Engineering College College of Science and Engineering Credit level (Normal year taken) SCQF Level 11 (Postgraduate) Availability Not available to visiting students SCQF Credits 10 ECTS Credits 5 Summary The Statistical Signal Processing course considers representing real-world signals by stochastic or random processes. The tools for analysing these random signals are developed in the Probability, Random Variables, and Estimation Theory course, and this course extends them to deal with time series. The notion of statistical quantities such as autocorrelation and auto-covariance are extended from random vectors to random processes, and a frequency-domain analysis framework is developed. This course also investigates the affect of systems and transformations on time-series, and how they can be used to help design powerful signal processing algorithms to achieve a particular task. The course introduces the notion of representing signals using parametric models; it extends the broad topic of statistical estimation theory covered in the Probability, Random Variables, and Estimation Theory course for determining optimal model parameters. In particular, the Bayesian paradigm for statistical parameter estimation is introduced. Emphasis is placed on relating these concepts to state-of-the-art applications and signals. This course, in combination with the Probability, Random Variables, and Estimation Theory course, provides the fundamental knowledge required for the advanced signal, image, and communication courses in the MSc course. Course description Any minor modifications to the latest syllabus and lectures are always contained in the lecture handout. Introduction and Review of Discrete-time Systems (3 lectures). 1. Introduction and course overview. Role of deterministic and random signals, and the various interpretations of random processes in the different physical sciences. 2. Brief review of Fourier transform theorem: a. Transforms for continuous-time, discrete-time, periodic or aperiodic, signals. b. Parseval�s Theorem. c. Properties of the discrete Fourier transform (DFT). d. The DFT as a linear transformation. e. Summary of frequently used transform pairs. 3. Review of discrete-time systems. a. Basic discrete-time signals. b. The z-transform and basic properties. c. Summary of frequently used transform pairs. d. Definitions of linear time-invariant (LTI) and linear time-varying (LTV) systems. e. Rational transfer functions; pole-zero models. f. Frequency response of LTI systems. g. Example of inverse bilateral z-transforms, and different approaches to get the same answer; partial fraction expansions using the cover-up rule. Stochastic Processes (6 lectures). 1. Introduction to stochastic processes, and their definition as an ensemble of deterministic realisations resulting from the outcome of a sample space; also covers the various interpretations of the samples of a random process. 2. Covers predictable processes with an example of harmonic processes; description of stochastic processes using probability density functions (pdfs). 3. Notion of stationary and nonstationary processes. 4. Statistical description of random processes; examples of some predictable processes through a MATLAB demonstration; second-order statistics including mean and autocorrelation sequences, with an example of calculating autocorrelation for a harmonic process. 5. Types of random processes, including independent, independent and identically distributed (i. i. d.) random processes, and uncorrelated and orthogonal processes. 6. Introduction to stationary processes, both order-N stationary, strict-sense stationary, and wide-sense stationary; example of testing whether a Wiener process is stationary or not; also covers wide sense periodic and wide-sense cyclo-stationary processes. 7. Notion of ergodicity, and the notion of time-averages being equal to ensemble averages in the mean-square sense. 8. Second-order statistical descriptions, including autocorrelation and covariances; joint-signal statistics; types of joint stochastic processes; correlation matrices. 9. Basic introduction to Markov processes. Frequency-Domain Description of Stationary Processes (3 lectures). 1. Introduction to random processes in the frequency domain, including the stochastic decomposition interpretation, the transform of averages interpretation, and the connections between these interpretations. 2. Formal definition of the power spectral density (PSD) and its properties; general form of the PSD including autocorrelation sequences (ACSs) with periodic components; the PSD of a harmonic signal (as a linear summation of sinusoids). 3. The PSD of common stationary processes: introducing white noise, harmonic processes, complex-exponentials. 4. Definition of the cross-power spectral density (CPSD), a physical overview, and the properties of the CPSD; an overview of complex spectral density functions, their relationships with PSDs, and how to find their inverses; properties of complex spectral densities. Linear systems with stationary random inputs (3 lectures). 1. Considers the effect of linear systems on random processes, and the resulting output processes; discusses the linearity of the expectation operator. 2. Develops the basic relationships between the input and output for stationary random processes, including input-output cross-correlation, output autocorrelation, and output power. Discusses the case of LTI systems, and the fact that most real world applications will be a LTV system. 3. System identification using cross-correlation. 4. Frequency-domain analysis of LTI systems, including input-output CPSD and output PSD. 5. Equivalence of time-domain and frequency-domain methods. 6. LTV systems with non-stationary inputs. Linear signal models (2 lectures). 1. Introduction to the notion of parametric modelling. 2. Nonparametric vs parametric signal models. 3. Types of pole-zero models. 4. All-pole models: impulse response, autocorrelation functions, poles, minimum-phase conditions. 5. Linear prediction, autoregressive (AR) processes, Yule-Walker equations. 6. All-zero models: impulse response, autocorrelation functions, zeros, and moving average (MA) processes. 7. Pole-Zero models: autocorrelation functions, autoregressive moving average (ARMA) processes. 8. Overview of extension to time-varying processes. 9. Applications and examples. Estimation Theory for Random Processes (3 lectures). 1. Sample autocorrelation and auto-covariance functions. 2. Least-squares for AR modelling. 3. Estimating signals in noise, using parametric signal models. 4. Bayesian estimation of sinusoids in noise, and other applications of Bayesian estimation methods to time-series analysis. 2 x 2 hour lectures, and 1 x 2 hour tutorial, per week from Week 6 to Week 11 (one lecture in Week 6 only).
 Pre-requisites Co-requisites Students MUST also take: Probability, Random Variables and Estimation Theory (PET) (MSc) (PGEE11123) Prohibited Combinations Other requirements None
 Academic year 2019/20, Not available to visiting students (SS1) Quota:  0 Course Start Block 2 (Sem 1) Timetable Timetable Learning and Teaching activities (Further Info) Total Hours: 100 ( Lecture Hours 44, Seminar/Tutorial Hours 22, Feedback/Feedforward Hours 11, Programme Level Learning and Teaching Hours 2, Directed Learning and Independent Learning Hours 21 ) Assessment (Further Info) Written Exam 100 %, Coursework 0 %, Practical Exam 0 % Additional Information (Assessment) 100% Examination Feedback Not entered No Exam Information
 At the end of the Statistical Signal Processing course, a student should be able to: 1. explain, describe, and understand the notion of a random process and statistical time series; 2. characterise random processes in terms of its statistical properties, including the notion of stationarity and ergodicity; 3. define, describe, and understand the notion of the power spectral density of stationary random processes; analyse and manipulate power spectral densities; 4. analyse in both time and frequency the affect of transformations and linear systems on random processes, both in terms of the density functions, and statistical moments; 5. explain the notion of parametric signal models, and describe common regression-based signal models in terms of its statistical characteristics, and in terms of its affect on random signals; 6. apply least squares, maximum-likelihood, and Bayesian estimators to model based signal processing problems.
 1. Recommended course text book: Therrien C. W. and M. Tummala, Probability and Random Processes for Electrical and Computer Engineers, Second edition, CRC Press, 2011. IDENTIFIERS -- Hardback, ISBN10: 1439826986, ISBN13: 978-1439826980 2. Manolakis D. G., V. K. Ingle, and S. M. Kogon, Statistical and Adaptive Signal Processing: Spectral Estimation, Signal Modeling, Adaptive Filtering and Array Processing, McGraw Hill, Inc., 2000. 3. Kay S. M., Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice-Hall, Inc., 1993. 4. Papoulis A. and S. Pillai, Probability, Random Variables, and Stochastic Processes, Fourth edition, McGraw Hill, Inc., 2002.
 Graduate Attributes and Skills Not entered Keywords Statistical Signal Processing,Discrete-time Random Signals,Linear Systems
 Course organiser Dr James Hopgood Tel: (0131 6)50 5571 Email: James.Hopgood@ed.ac.uk Course secretary Mrs Megan Inch-Kellingray Tel: (0131 6)51 7079 Email: M.Inch@ed.ac.uk
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