Continuous-Time Signals

Welcome to the continuous-time signals section of this course.

What is a Signal?

A signal is a function that conveys information about a phenomenon. In its most general form, a signal is a mapping from an independent variable (often time or space) to a dependent variable that represents some physical quantity.

Definition: A signal is a function $x:\mathcal{D}\to \mathcal{R}$ where $\mathcal{D}$ is the domain (e.g., time) and $\mathcal{R}$ is the range (e.g., amplitude).

In this course, we focus on one-dimensional signals where the independent variable is time. Such signals are ubiquitous in engineering and science:

• Audio signals: sound pressure variations over time
• Biomedical signals: ECG (heart activity), EEG (brain activity)
• Communication signals: voltage or current in electronic circuits
• Financial signals: stock prices, economic indicators
• Environmental signals: temperature, pressure, seismic activity

Signal Classification

Signals can be classified based on different characteristics.

Continuous-Time vs Discrete-Time Signals

Continuous-time signals are defined for all values of time $t\in \mathbb{R}$. We denote them as $x(t)$.

Discrete-time signals are defined only at discrete instants, typically integer multiples of a sampling period. We denote them as $x[n]$ where $n\in \mathbb{Z}$.

Deterministic vs Random Signals

• Deterministic signals: can be described by an explicit mathematical expression
• Random (stochastic) signals: cannot be predicted exactly; described statistically using a probability density function.

Energy and Power Signals

The energy of a continuous-time signal is:

$$E_x={\int }_{-\mathrm{\infty }}^{+\mathrm{\infty }}|x(t){|}^{2}dt$$

The average power is:

$$P_x=\underset{T\to \mathrm{\infty }}{lim}\frac{1}{2T}{\int }_{-T}^T|x(t){|}^{2}dt$$

• Energy signal: $0<E_x<\mathrm{\infty }$ (and $P_x=0$)
• Power signal: $0<P_x<\mathrm{\infty }$ (and $E_x=\mathrm{\infty }$)

What is a System ?

A system is a mathematical model that transforms an input signal into an output signal. Systems can be classified based on the number of inputs and outputs:

SISO Systems

A Single-Input Single-Output (SISO) system has one input signal $x(t)$ and one output signal $y(t)$:

$$y(t)=\mathcal{T}\{x(t)\}$$

MIMO Systems

A Multiple-Input Multiple-Output (MIMO) system has multiple inputs $x_1(t),x_2(t),\dots ,x_n(t)$ and multiple outputs $y_1(t),y_2(t),\dots ,y_m(t)$. MIMO systems are described using matrix representations.

Objectives

This module covers the fundamental concepts of analog signal processing:

• Representation and classification of continuous-time signals
• Analysis of SISO Linear Time-Invariant (LTI) systems
• Fourier series decomposition of periodic signals
• Fourier transform for aperiodic signals
• Laplace transform and its applications

Prerequisites

• Calculus (integrals, derivatives)
• Complex numbers
• Basic linear algebra

Notation

A continuous-time signal is denoted $x(t)$ where $t\in \mathbb{R}$ represents time.