Signals: Dimensionality, Time, and Signal Types

• A signal is a function of one or more variables conveying information about a physical phenomenon.

Dimensionality of Signals

• Signals are classified based on the number of independent variables.
• A one-dimensional (1D) signal is a function of only one variable.
• Human speech, represented fluctuations in air pressure over time is a 1D signal.
• A multidimensional signal is a function of two or more variables.
• An example of a 2D signal is a monochromatic image, which is a measure of light intensity as a function of horizontal and vertical displacement.

Continuous-Time and Discrete-Time Signals

• Signals are classified as continuous-time or discrete-time, based on whether they are functions of continuous or discrete variables.
• A continuous-time signal is a function of continuous variables (e.g., a real variable).
• A discrete-time signal is a function of discrete variables (e.g., an integer variable).
• Digital images are discrete-time signals, where variables correspond to horizontal and vertical position.
• A digital signal is a discrete-time signal where the value of the function is also discrete.
• An analog signal is a signal with continuous variables and a continuous function value.
• Many physical world phenomena are continuous-time signals, examples include voltage, current waveforms, electrocardiograms, position/velocity/acceleration of a moving body and flow rates of liquids or gases.
• Signals processed by digital computers are discrete-time signals, examples include digital video, digital photographs, and digital audio data.
• A discrete-time signal may be inherently discrete or sampled from a continuous-time signal.
• The Dow Jones Industrial Average is an inherently discrete signal defined at daily intervals.
• A sampled version of a speech signal would be an example of discrete-time signal sampled from a continuous-time signal.

Notation and Graphical Representation of Signals

• Continuous-time signals are written with independent variables in parentheses, e.g., x(t).
• Discrete-time signals are written with independent variables in brackets, e.g., x[n].
• Discrete-time signals are sometimes referred to as sequences.

Examples of Signals

• Digitized speech signal and monochromatic image are examples of graphical representations of signals.

Systems

• A system processes one or more input signals to produce one or more output signals.
• A system can be represented mathematically by equations.
• In a communication system, the input could be the message to send, and the output could be the received message.
• In robotics, the input might be the desired position, and the output could represent the actual position.

Classification of Systems

• Systems are classified based on the number of inputs and outputs.
• A single-input system has only one input.
• A multi-input system has multiple inputs.
• A single-output system has only one output.
• A multi-output system has multiple outputs.
• Common system types are single-input single-output (SISO) and multi-input multi-output (MIMO).
• A system that deals with continuous-time signals is a continuous-time system.
• A system that deals with discrete-time signals is a discrete-time system.
• A system handling both continuous- and discrete-time signals is a hybrid or sampled-data system.
• Systems that deal with digital signals are digital systems, while systems that handle analog signals are analog systems.
• Systems are classified as 1D or multidimensional based on whether they interact with 1D or multidimensional signals.

Examples of Systems

• Systems can extract information, like speaker identification from speech signals or detecting heart abnormalities from electrocardiograms.
• Amplification and noise reduction are functionalities that systems offer.
• An RC network is a basic system where the input is the source voltage and the output is the capacitor voltage.
• A communication system takes a message at one location and reproduces it at another location.
• A control system tracks a reference input, like in robotics where the output (actual position) follows the reference input (desired position).

Continuous-Time Signals and Systems

• The focus of the course is 1D continuous-time signals and systems, mostly single-input single-output (SISO) systems.
• Discrete-time and multidimensional cases are treated in other courses.

Why Study Signals and Systems?

• There are practical situations needing to develop systems that manipulate/process signals.
• A formal mathematical framework is needed to study such systems.
• The goal of the course is to provide students with this framework.