Discrete-time signals

Introduction

Screencast video [⯈]

Module overview

This module will cover some very basic characteristics and examples of signals in the discrete-time domain.

1. Sampling process - In the discrete domain the signals are represented as sequences of numbers called samples.
2. Elementary signals - All discrete-time signals can be written as a set of elementary signals, therefore a good understanding of these elementary signals helps with the analysis of more complex signals.
3. Signal duration - Discrete-time signals may be conveniently classified in terms of their duration or extent.
4. Periodicity - A discrete-time signal may always be classified as either periodic or aperiodic, which refers to whether the signal repeats itself.
5. Symmetry - A discrete-time signal will often possess some form of symmetry that may be exploited in solving problems.
6. Signal manipulations - Signal manipulations are generally decomposed of a few basic transformations.

Summary

Elementary signals

Signal duration

Signal properties

$$\begin{array}{ccc}\text{Periodic}& :& x\left[n\right]=x[n+N]\\ \text{Even}\text{Conjugate-symmetric}& :& x\left[n\right]=x[-n]x\left[n\right]=x^{\ast }[-n]\\ \text{Odd}\text{Conjugate-antisymmetric}& :& x\left[n\right]=-x[-n]x\left[n\right]=-x^{\ast }[-n]\\ \text{General:}x\left[n\right]& =& x_e\left[n\right]+x_o\left[n\right]\\ \text{with}{x}_e\left[n\right]& =& \frac{1}{2}\left(x\right[n]+x[-n\left]\right)\\ \text{and}{x}_o\left[n\right]& =& \frac{1}{2}\left(x\right[n]-x[-n\left]\right)\end{array}$$

Signal manipulations

Amplitude transformations: $$\begin{array}{ccc}\text{Addition}& :& y\left[n\right]=x_1\left[n\right]+x_2\left[n\right]\\ \text{Multiplication}& :& y\left[n\right]=x_1\left[n\right]\cdot x_2\left[n\right]\\ \text{Scaling}& :& y\left[n\right]=c\cdot x\left[n\right]\end{array}$$

Transformation of independent variable $n$:

$$\begin{array}{ccc}\text{Time shifting (delay or advance)}& :& f\left[n\right]=n-n_0\\ \text{Time reversal}& :& f\left[n\right]=-n\\ \text{Time scaling = Down- or Up- sampling}& :& f\left[n\right]=M\cdot n\text{or}f\left[n\right]=n/M\end{array}$$ Note: Shifting, reversal, and time scaling operations are order dependent.