Discrete-Time Signals

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

Objectives

This module covers the fundamental concepts of digital signal processing:

• Representation and classification of discrete-time signals
• Analysis of discrete LTI systems
• Discrete-Time Fourier Transform (DTFT)
• Z-transform and its applications
• Sampling theory and reconstruction

Prerequisites

• Continuous-time signal analysis
• Numerical sequences
• Complex numbers

Notation

A discrete-time signal is denoted $x[n]$ where $n\in \mathbb{Z}$ is the discrete time index.

The energy of a discrete signal is:

$$E_x=\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}|x[n]{|}^2$$

The average power of a discrete signal is:

$$P_x=\underset{N\to \mathrm{\infty }}{lim}\frac{1}{2N+1}\sum _{n=-N}^N|x[n]{|}^2$$

Relationship with Continuous Signals

A discrete signal can be obtained by sampling a continuous signal:

$$x[n]=x_c(n{T}_s)$$

where $T_s$ is the sampling period and $f_s=1/T_s$ is the sampling frequency.