Continuous vs Discrete Signals

Introduction

The distinction between continuous-time and discrete-time signals is the most fundamental classification in signal processing. A continuous-time signal is defined for every real-valued time instant — it exists at $t = 1.5$ seconds, at $t = \pi$ seconds, and at every point in between. A discrete-time signal exists only at isolated integer indices — it has values at $n = 0, 1, 2, 3, \ldots$ but is literally undefined between these points.

This distinction is not merely mathematical — it reflects the physical difference between analog phenomena (voltage varying smoothly in a wire, pressure changing continuously in air) and digital representations (numbers stored in computer memory at discrete time steps). Understanding both signal types and how they relate through sampling and reconstruction is essential because modern engineering systems invariably bridge both worlds.

Continuous-Time Signals

Definition

A continuous-time signal $x(t)$ is a function defined for all real values of $t \in (-\infty, +\infty)$. At any arbitrary time instant — whether rational, irrational, or transcendental — the signal has a well-defined value.

Representation

Continuous-time signals are typically represented by:

• Mathematical functions: $x(t) = A\sin(2\pi f_0 t + \phi)$, $x(t) = e^{-at}u(t)$
• Graphs: smooth curves with time on the horizontal axis
• Differential equations: $\frac{dy}{dt} + 2y = x(t)$ relates input to output

Physical Examples

• Voltage across a resistor: varies smoothly as current changes
• Acoustic pressure wave: air pressure oscillates continuously
• Temperature: changes gradually over time
• Electromagnetic field: varies continuously in space and time
• Human speech: pressure variations produced by vocal cords

Mathematical Operations

Continuous-time signals support operations involving calculus:

• Differentiation: $\frac{dx}{dt}$ (rate of change)
• Integration: $\int x(\tau)d\tau$ (accumulation)
• Continuous convolution: $\int x(\tau)h(t-\tau)d\tau$

Discrete-Time Signals

Definition

A discrete-time signal $x[n]$ is a sequence of numbers defined only at integer indices $n \in \{\ldots, -2, -1, 0, 1, 2, \ldots\}$. Between indices, the signal simply does not exist — it is not zero; it is undefined.

Representation

Discrete-time signals are represented by:

• Sequences: $x[n] = \{\ldots, 0, 1, 3, 2, 0, \ldots\}$ (with an arrow marking $n=0$)
• Formulas: $x[n] = a^n u[n]$, $x[n] = \cos(\Omega_0 n)$
• Stem plots: vertical lines at integer positions (NOT connected by curves)
• Difference equations: $y[n] = 0.5y[n-1] + x[n]$

Physical Examples

• Sampled audio (CD): 44,100 numbers per second representing sound
• Stock prices: daily closing prices form a discrete sequence
• Digital sensor readings: temperature measured every minute
• Image pixels: spatial samples of continuous light patterns
• Daily rainfall: accumulated rainfall recorded once per day

Mathematical Operations

Discrete-time signals use summation and differences:

• First difference: $x[n] - x[n-1]$ (discrete derivative)
• Running sum: $\sum_{k=-\infty}^n x[k]$ (discrete integration)
• Discrete convolution: $\sum_k x[k]h[n-k]$

Key Differences

The Sampling Connection

Discrete-time signals most commonly arise by sampling continuous-time signals:

$$x[n] = x_c(nT_s)$$

where $T_s$ is the sampling period and $f_s = 1/T_s$ is the sampling rate. The Nyquist-Shannon theorem guarantees that if $f_s > 2f_{max}$, the continuous signal can be perfectly reconstructed from the discrete samples.

Frequency Relationship

The continuous frequency $\omega$ (rad/s) and discrete frequency $\Omega$ (rad/sample) are related by: $$\Omega = \omega T_s = \frac{2\pi f}{f_s}$$

The continuous frequency range $[0, f_s/2]$ maps to the discrete range $[0, \pi]$.

Periodicity Differences

A continuous-time signal $x(t) = \cos(\omega_0 t)$ is always periodic with period $T = 2\pi/\omega_0$.

A discrete-time signal $x[n] = \cos(\Omega_0 n)$ is periodic only if $\Omega_0/(2\pi)$ is a rational number. If irrational, the sequence never exactly repeats!

Example: $x[n] = \cos(0.5\pi n)$ has period $N = 4$ ($\Omega_0/2\pi = 1/4$ is rational).

But $x[n] = \cos(n)$ ($\Omega_0 = 1$ rad, $\Omega_0/2\pi = 1/2\pi$ is irrational) is NOT periodic as a discrete sequence.

Practical Implications

Why we use both: The physical world is continuous (analog), but processing and storage are discrete (digital). Modern systems sample analog signals, process them digitally, and reconstruct analog outputs.

Analog advantages: No quantization noise, no aliasing, infinite resolution (theoretically)

Digital advantages: Perfect reproduction (copying without loss), programmable processing, noise immunity, compact storage, easy transmission

The ADC-DSP-DAC chain: Analog → ADC (sampling + quantization) → Digital Processing → DAC (reconstruction) → Analog

Key Takeaways

• Continuous-time signals exist for all real $t$; discrete-time signals exist only at integer $n$
• Continuous signals are processed with calculus (integration, differentiation); discrete with summation and differences
• Sampling bridges the two worlds: $x[n] = x_c(nT_s)$, requiring $f_s > 2f_{max}$
• Discrete frequencies are periodic (range $[0, 2\pi]$); continuous frequencies are not
• Discrete periodicity requires rational frequency ratio — unlike continuous-time
• Modern systems combine both: analog sensing → digital processing → analog output