Chapter 1 - What is a signal?

Time-like Signals

Most of the signals we encounter when studying signals and systems are time-like, so we'll formalise this mathematically.

Continuous-time - Real-valued Signals

Most signals we encounter are real-valued, so we can define them as:

\[ x:\quad \mathbb{R}\to \mathbb{R},\quad t\mapsto x(t) \]

Continuous-time - Complex-valued Signals

However, some signals will actually be complex:

\[ z:\quad \mathbb{R}\to \mathbb{C},\quad t\mapsto z(t) \]

Discrete-time - Real-valued Signals

We can also have discrete-time signals

\[ x:\quad \mathbb{Z}\to \mathbb{R},\quad n\mapsto x[n] \]

Discrete-time - Complex-valued Signals

We can also have discrete-time complex-valued signals

\[ z:\quad \mathbb{Z}\to \mathbb{C},\quad n\mapsto z[n] \]

Discrete-time - Quantized Signals

If the output is restricted to only discrete values, i.e., through quantization/digitization, we can map each discrete value to an integer:

\[ x_{q}:\quad \mathbb{Z}\to \mathbb{Z},\quad n\mapsto x_{q}[n] \]

Deterministic vs Stochastic (Random)

A deterministic signal is a signal can be predict exactly by a mathematical formula. There is no uncertainty or randomness for a deterministic signal.

This is the simplest model, and while it's ideal, it can still be quite good at representing reality.

As a simple example, we can model the signal of a simple harmonic oscillator as: $$ x(t) = A\sin{(2\pi f_{0} t + \phi)} $$

On the other hand, a non-deterministic signal, also known as a stochastic signal has an element of randomness. While the exact value of a stochastic signal cannot be predicted beforehand, we can model the statistical properties of the signal.

This is more suited in modelling the behaviour of real-world systems, but is an added layer of complexity that isn't strictly necessary for all applications.

An example of this would be a simple voltage measurement of current for a noisy source, due to thermal noise. $$ i(t) = i_{s} + i_{\text{noise}}(t) $$ $$ i_{\text{noise}}(t)\sim\mathcal{N}(0,\sigma^{2}) $$ Here we have the deterministic part \(i_{s}\), which in this case is constant.

And we have the non-deterministic part \(i_{\text{noise}}(t)\), which is modelled as additive white gaussian noise (AWGN).

While we don't know the actual value of this signal, we know that the average value will be \(i_{s}\).

Next Chapter: Signal Properties

In the next chapter, we'll explore some of the properties of signals.