EE313 Linear Systems and Signals will help you build a mathematical foundation for analyzing signals and systems in a wide variety of applications, including speech, audio, image and video processing as well as communications and control systems. Signals are functions of time, e.g. temperature measured over time, and can be represented as data values or abstractly as mathematical formulas or properties. A system converts an input signal into an output signal. You'll analyze signals and systems in continuous-time and discrete-time using mathematical theory and computer simulation. Analysis of signals and systems in time, frequency and generalized frequency domains requires quantitative reasoning. The quantitative reasoning involves mathematical modeling of signals and systems as well as applying transforms to them involving integration or summation. The course covers representation of signals and systems, system properties, sampling, Laplace and z-transforms, transfer functions, frequency responses, convolution, stability, Fourier transforms, feedback, and control applications. You'll simulate signals and systems by computer using MATLAB.

*Sinusoidal signals*: A sinusoid can represent a note on the piano keyboard;
e.g., middle 'A' has a principal frequency of 440 Hz which can be represented as
cos(2 pi f_{0} t) where f_{0} = 440 Hz.
A sinusoid can also represent
a radio station frequency (e.g. *f*_{0} = 90.5 MHz for KUT),
Wi-Fi frequency (e.g. *f*_{0} = 2.432 GHz for channel 9 in the 2.4
GHz band) or a GPS frequency (e.g. *f*_{0} = 1.55 GHz).

*Analysis of periodic signals*: A periodic signal is one that repeats.
A sinusoidal signal is an example.
You'll learn to decompose an existing periodic signal into a sum of
sinusoids using Fourier series.
An example is an A major chord consisting of notes A, C# and E which
correspond to frequencies
*f*_{A} = 440 Hz,
*f*_{C#} = 550 Hz and
*f*_{E} = 660 Hz,
respectively, based on the note A being in the fourth octave.
Speech sounds can be similarly decomposed.

*Synthesis of periodic signals*: A periodic signal can be create a periodic
signal by adding sinusoidal signals together.
For example, one could play single notes e.g. cos(2 pi *f*_{A} *t*)
or a chord by playing notes at the same time
e.g. cos(2 pi *f*_{A} *t*) +
cos(2 pi *f*_{C#} *t*) +
cos(2 pi *f*_{E} *t*).
One can generate more general periodic signals using Fourier series.
Synthesis of periodic signals is useful in synthesizing speech sounds by
computer or smart phone.
A smart phone actually synthesizes the speech heard in a phone call.

*Filtering*: A filter is a type of system that tries to change the content
of the input signal to produce an output signal.
An example is an equalizer in audio system in which the lowest sub-woofer
frequencies, mid-range woofer frequencies and high tweeter frequencies can
be amplified by different values.
This allows one to pump up the bass or make the bass quieter.
Filters are generally designed in the generalized frequency domain
(Laplace or z domains) and analyzed in both the time domain using
convolution and frequency domain using Fourier transforms.

*Feedback*. A feedback system feed its output signal back into the system.
The course discusses a type of feedback system called an infinite impulse
response (IIR) filter.
An IIR filter generally requires fewer arithmetic computations to realize
the same frequency response than a filter without feedback.
However, IIR filters may create unreliable output if not designed carefully.
You'll learn how to design IIR filters in the generalized frequency domain,
analyze them in the time and frequency domains, and simulate them by
filtering practical signals such as audio signals.

*Abstraction*. The textbook and lecture material guide you through
several stages of abstraction.
You'll take a significant jump in abstraction to generalize the
notion of continuous-time systems beyond circuits and continuous-time
signals beyond voltage and current values in the pre-requisite electrical
and computer engineering courses.
You'll then take another jump in abstraction to discrete-time
systems and signals, in which physical time measured in seconds is
abstracted away.
You'll then convert continuous-time and discrete-time signals and
systems into equivalent frequency-domain representations using integration
and summation operations.
Finally, you'll generalize the frequency-domain representations also
using integration and summation operations.

*Mathematical Modeling*. Through in-lecture exercises and homework,
you'll develop the skills to rigorously tackle word problems,
i.e., questions that are ill-specified or not directly stated in
mathematical form and that need to be appropriately modeled and/or
mapped to a mathematical setting to enable rigorous assessment.

*Intuition*. One can approach the analysis in any combination
of the following three domains: time, frequency, or generalized
frequency.
All three domains provide insight, and you'll gain experience about
which ones to use to simplify the quantitative reasoning.
You'll also gain the ability to move among the three representations
using the appropriate mathematical transformations.
A key part in building intuition is in visualizing signals and systems
in the three domains as well as the relationships among the domains,
which the you'll gain from hand sketches, textbook illustrations,
online demos, and your own computer simulations.

Last updated 08/21/23. Send comments to bevans@ece.utexas.edu