# ELEC441 Linear Systems and Noise

## Paper Description

Introduces the "systems" approach to solving physical problems. In this approach devices or other physical systems are treated as black boxes that convert inputs signals to output signals. Covers generalized functions, classification of systems, the Fourier transform, sampling and the FFT, causality and the Kramers Kronig relations, noise processes and matched filtering.

To give students familiarity with the "systems" approach to analyzing physical problems. As well as giving the students analytic and computational skills, the course aims do help develop their physical intuition.

This paper consists of 15 lectures and 6 tutorials. There are 3 assignments.

Assesment:
Final Exam 70%, Assignments 30%

Important information about assessment for ELEC441

Course Coordinator:

After completing this paper students are expected to have achieved the following major learning objectives:

• Have a good understanding of the delta function and generalized functions in general. Be able to use the formal definition of generalized functions for doing calculus on generalized functions.
• Understand the convolution integral and it's relation to the delta function and the superposition principle.
• Be familiar with the Fourier transform and its properties. Be comfortable finding Fourier transforms using the properties of the Fourier transform and the Fourier transforms for a base set of functions.
• Find the Fourier transform of generalized functions from the definition.
• Understand sampling and its effects in the Fourier domain. Be able to derive the sampling theorem and show the relationship between the discrete and continuous Fourier transforms.
• Understand the effect of causality on a system transfer function, the Hilbert transform and the Kramers-Kronig relation.
• Be able to solve problems related to the one dimensional propagation of a signal through a dispersive and for the narrow bandwidth approximation derive expressions for the group and phase velocities.
• Be introduced to stationary stochastic processes, and be able to calculate the effect of a linear system on the power spectrum of a signal.
• Be able to use matched filtering to optimally find signals in noise.

Topics:

• The Dirac delta-function
• Generalised functions and functions, Linear systems
• Classes of systems. Convolution
• Stability of linear time invariant systems, eigenfunctions of linear time invariant systems, cascaded systems
• The Fourier transform and its inverse
• Examples of Fourier transforms, convergence of Dirichlet integrals
• Fourier transform of generalised functions
• Sampling in frequency and time
• The sampling theorem, Bernstein's theorem
• Discrete Fourier transform
• The fast Fourier transform (FFT)
• The Hilbert transform, causality and Kramers-Kronig relations, 1D propagation in a dispersive medium
• Energy power in deterministic signals
• Stochastic process and noise, power spectrum of stationary stochastic processes
• Signals in noise

Resources:
The course will closely cover the notes "Linear systems and Noise" by Tan and Fox.

Some books that cover similar material and are available in the library are:

• The Fourier transform and its applications, R. N. Bracewell.
• The Fourier integral and its applications, A. Papoulis.
• Probability, random variables, and stochastic processes, A. Papoulis.
• Noise, F. R. Connor.
• Signals and Systems, A. V. Oppenheim and A. S. Willsky.

# Formal University Information

The following information is from the University’s corporate web site.

## Details

An introduction to the "systems" approach to solving physical problems: generalised functions, the Fourier transform, sampling and the FFT, causality and the Kramers-Kronig relations, noise processes and matched filtering.

Paper title Linear Systems and Noise ELEC441 Electronics 0.0833 10 points First Semester \$666.57 \$2,895.09
Limited to
BSc(Hons), PGDipSci, MSc, MAppSc
Contact

Teaching staff

Textbooks

Textbooks are not required for this paper.

Global perspective, Interdisciplinary perspective, Lifelong learning, Scholarship, Communication, Critical thinking, Information literacy, Self-motivation, Teamwork.
Learning Outcomes

After completing this paper students are expected to:

1. Have a good understanding of the delta function and generalised functions in general and be able to use the formal definition of generalised functions for doing calculus on generalised functions.
2. Understand the convolution integral and its relation to the delta function and the superposition principle.
3. Be familiar with the Fourier transform and its properties and be comfortable finding Fourier transforms using the properties of the Fourier transform and the Fourier transforms for a base set of functions.
4. Find the Fourier transform of generalised functions from the definition.
5. Understand sampling and its effects in the Fourier domain and be able to derive the sampling theorem and show the relationship between the discrete and continuous Fourier transforms.
6. Understand the effect of causality on a system transfer function, the Hilbert transform and the Kramers-Kronig relation.
7. Be able to solve problems related to the one dimensional propagation of a signal through a dispersive and for the narrow bandwidth approximation derive expressions for the group and phase velocities.
8. Be introduced to stationary stochastic processes and be able to calculate the effect of a linear system on the power spectrum of a signal.
9. Be able to use matched filtering to optimally find signals in noise.

## Timetable

### First Semester

Location
Dunedin
Teaching method
This paper is taught On Campus
Learning management system
None