Introduction and motivation

Introduction and motivation

Most areas of electrical and computer engineering (beyond signal processing) deal with signals. Communications is about transmitting, receiving, and interpreting signals. Signals are used to probe and model systems in control and circuit design. The images acquired by radar systems and biomedical devices are signals that change in space and time, respectively. Signals are used in microelectronic devices to convey digital information or send instructions to processors.

This course will provide a mathematical framework to handle signals and operations on signals. Some of the questions that will be answered in this course include:

• What is a signal? How do we represent it?
• How do we represent operations on signals?
• What does it mean for signals to be similar/different from each other?
• When is a candidate signal a good/bad approximation (i.e., a simplified version) of a target signal?
• When is a signal “interesting” or “boring”?
• How can we characterize groups of signals?
• How do we find the best approximation of a target signal in a group of candidates?

Course overview

Signal theory

The signal theory presented in this course has three main components:

• Signal representations and signal spaces , which provide a framework to talk about sets of signal and to define signal approximations.
• Distances and norms to evaluate and compare signals. Norms provide a measure of strength, amplitude, or “interestingness” of a signal, and distances provide a measure of similarity between signals.
• Projection theory and signal estimation to work with signals that have been distorted, aiming to recover the best approximation in a defined set.

Operator theory

Operators are mathematical representations of systems that manipulate a signal. The operator theory presented in this course has three main components:

• Operator properties that allow us to characterize their effect on signals in a simple fashion.
• Operator characterization that allow us to model their effect on arbitrary inputs.
• Operator operations (no pun intended) that allow us to create new systems and reverse the effect of a system on a signal.

Optimization theory

Optimization is an area of applied mathematics that, in the context of our course, will allow us to determine the best signal output for a given problem using defined metrics, such as signal denoising or compression, codebook design, and radar pulse shaping. The optimization theory presented in this course has three main components:

• Optimization guarantees that rely on properties of the metrics and signal sets we search over to formally ensure that the optimal signal can be found.
• Unconstrained optimization , where we search for the optimum over an entire signal space.
• Constrained optimization , where the optimal signal must meet additional specific requirements.

Example

As an example, consider the following communications channel:

A mathematical formulation of this channel requires us to:

• establish which signals $x$ can be input into the transmitter;
• how the transmitter $F$ , the channel $H$ , and the receiver $G$ are characterized;
• how the concatenation of the blocks $F$ and $H$ is expressed;
• how the noise addition operation is formulated;
• how we measure whether the decoded message $\widehat{x}$ is a good approximation of the input $x$ ;
• how is the receiver $G$ designed to be optimal for all the choices above.

For this example, by the end of the course, you will be able to solve the problem of selecting the transmitter/receiver pair $F,G$ that minimizes the power of the error $e=\widehat{x}-x$ while meeting maximum transmission power constraints $\frac{\mathrm{power}\left(F\right(x\left)\right)}{\mathrm{power}\left(x\right)}<P_{max}$ .