# TSDT14 Signal Theory

TSDT14 Signal Theory is a course in signal processing, dealing with both time-continuous and time-discrete signals, that are both deterministic and stochastic. The focus is on stochastic signals. The signal processing systems are usually linear and time-invariant, but we also consider some momentary non-linear systems. One part of the course deals with multi-dimensional processes, where we concentrate on two-dimensional processes. We also consider traditional analog modulation methods and perform noise analysis of them. We are especially interested in transformations between time-continuous and time-discrete signals. A smaller part of the course is estimation of power spectral densities using Fourier methods, where DFT is used. Signal Theory is a basis for further studies in signal processing, e.g. Tele(data)-communication, image processing, automatic control, et. cetera. ## Course topics

• Time continuous and time discrete, real as well as complex, stochastic processes: Probability distribution, probability density, expectation, ensemble expectation, auto correlation function, power spectral density, cross correlation function, cross spectral density, stationarity, ergodicity. Especially Gaussian processes and white processes. Multidimensional processes.

• LTI filtering of stochastic processes: Relations between statistical properties of the input process and the output process. Especially matched filters and white Gaussian noise as input.

• Amplitude and angle modulation of stochastic processes: Relations between statistical properties of the input process and the output process. Especially Gaussian processes as input. Noise analysis of those modulation forms, primarily with white Gaussian noise as disturbance.

• Non-linear momentary systems: Quantization and monomial non-linearities. Relations between statistical properties of the input process and the output process. Especially Gaussian processes as input. Properties of quantization noise.

• Transformation between time continuous and time discrete stochastic processes: Sampling and pulse amplitude modulation, the sampling theorem, reconstruction and reconstruction error.

• Case study: Reconstruction in CD players.

• Estimation of expectations, auto correlation function and power spectral density.

## Course material

• The course is based on the book:
• Mikael Olofsson, Signal Theory, Studentlitteratur, 2011.
• Supplementary material:
• Mikael Olofsson, Tables and Formulas for Signal Theory, Studentlitteratur, 2011.
• Lab Memo: Mikael Olofsson, Time-Discrete Stochastic Signals, August 2017.
• Additional lab material: Mikael Olofsson, Short Matlab Manual, August 2017.
• Mikael Olofsson, Signal Theory – Additional Material on Complex Signals.

## Prerequisites

• From Calculus: Derivatives and integrals.
• From Probability theory: Most, but with focus on binary distributions, rectangular distributions and Gaussian distributions.
• From Signals and systems or Linear Systems: Fourier transforms, LTI systems, convolution, amplitude modulation, sampling and pulse amplitude modulation. (all of it deterministic)

## Information for enrolled students

For detailed lecture, tutorial and lab plans, see the course room in