What Is a Signal? Continuous vs Discrete-Time
The definition of a signal, and the fundamental distinction between continuous-time and discrete-time representations
Every voltage you measure, every sound you hear, every bit your computer processes — each is a signal. But what exactly makes something a signal, and why do engineers split them into two fundamentally different families?
Defining a Signal
A signal, in its most general sense, is a function that conveys information about some phenomenon. In electrical engineering, that phenomenon is most often a physical quantity — a voltage, a current, a sound pressure, an electromagnetic field strength — that changes over time. The key idea is that the variation itself carries meaning: a constant voltage tells you very little, but a voltage that rises and falls in a particular pattern can encode speech, transmit data, or indicate a sensor reading.
Mathematically, we represent a signal as a function of an independent variable. In this course, that independent variable is almost always time, denoted t. The signal's value at any given time is the output of the function at that input. This functional viewpoint is the foundation of everything that follows in the signals-and-systems portion of the course: a signal is a function, and studying signals means studying the properties and behaviors of functions.
Signals need not be functions of time — an image, for example, is a function of two spatial variables. However, the vast majority of this course deals with one-dimensional signals that depend on time, and we will use time as our default independent variable throughout.
Continuous-Time Signals
The first major category is the continuous-time signal. A continuous-time signal is defined for every real value of the independent variable t. If you pick any real number, the signal x(t) has a well-defined value at that instant — no gaps, no missing points.
We write a continuous-time signal using parentheses: x(t). This notation is deliberate and maintained throughout the course: the parentheses signal that t ranges over the entire real line.
Physically, continuous-time signals arise naturally in the world around us. The voltage produced by a microphone, the current through an inductor, the electric field radiated by an antenna — all are continuous-time signals because the underlying physical quantities exist and vary smoothly at every instant. In the language of analog electronics, these are analog signals.
Discrete-Time Signals
The second major category is the discrete-time signal, defined only at specific, separated instants of time — most commonly, at integer multiples of a fixed sampling interval T. Between those instants, the signal simply has no defined value.
We write a discrete-time signal using square brackets: x[n]. The index n is an integer (…, −2, −1, 0, 1, 2, …), serving as the independent variable in place of the continuous variable t. The brackets immediately tell the reader: this signal lives on integers, not the real line.
Discrete-time signals most commonly arise when we sample a continuous-time signal. If x(t) is a continuous-time signal and we sample it at intervals of T seconds, the resulting discrete-time signal is x[n] = x(nT). This sampling operation is the bridge between the analog world — where signals exist at every instant — and the digital world, where a computer or DSP can only store and manipulate a finite set of numbers at discrete indices.
This notational distinction is not cosmetic — it is a load-bearing convention used throughout the entire course. Whenever you see x(t) with parentheses, think real-valued time, every instant. Whenever you see x[n] with brackets, think integer index, samples only. Mixing up the two will lead to errors in convolution, Fourier analysis, and system properties later on.
Why Study Both?
Modern engineering systems rarely live entirely in one domain. A typical signal-processing chain begins with a continuous-time signal from the physical world. An analog-to-digital converter (ADC) samples this signal at regular intervals, producing a discrete-time sequence that a microcontroller or DSP can process. The result may then be converted back to a continuous-time signal by a digital-to-analog converter (DAC) to drive a speaker, motor, or display.
Because real systems constantly cross the boundary between continuous and discrete time, this course treats both domains in parallel. When we introduce a concept — linearity, time-invariance, convolution, Fourier analysis — we will examine it in both continuous and discrete time. The mathematical tools differ in important details, but the underlying intuitions transfer.
- Substitute t = 3 into the expression: x(3) = 2(3) + 1.
- Evaluate: x(3) = 6 + 1 = 7.
- The signal value at t = 3 s is 7 V.
- The index n = 3 corresponds to continuous time t = nT = 3 × 0.5 = 1.5 s.
- Evaluate the continuous-time signal: x(1.5) = 2(1.5) + 1 = 3 + 1 = 4.
- Therefore x[3] = x(1.5) = 4.
Check your understanding
- A signal is a function that conveys information, typically varying with time — common examples include voltage, current, and sound pressure.
- Continuous-time signals x(t) are defined for every real value of t and use parentheses; they correspond to analog physical quantities.
- Discrete-time signals x[n] are defined only at integer indices n and use square brackets; they typically arise from sampling a continuous-time signal.
- The sampling relationship x[n] = x(nT) bridges the continuous and discrete domains, where T is the sampling interval.