Introduction
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Module overview
This module will cover some very basic characteristics and examples of signals in the discrete-time domain.
- Sampling process - In the discrete domain the signals are represented as sequences of numbers called samples.
- 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.
- Signal duration - Discrete-time signals may be conveniently classified in terms of their duration or extent.
- Periodicity - A discrete-time signal may always be classified as either periodic or aperiodic, which refers to whether the signal repeats itself.
- Symmetry - A discrete-time signal will often possess some form of symmetry that may be exploited in solving problems.
- Signal manipulations - Signal manipulations are generally decomposed of a few basic transformations.
Summary
Elementary signals
| Name | Description | Remark |
|---|---|---|
| Delta pulse | $$ \delta[n] = \begin{cases} 1 & \text{for } n=0 \newline 0 & \text{otherwise} \end{cases} $$ | $$ u[n] - u[n-1] $$ |
| Unit step | $$ u[n] = \begin{cases} 1 & \text{for } n\geq 0 \newline 0 & \text{otherwise} \end{cases} $$ | $$ \sum_{k=0}^\infty \delta[n-k] $$ |
| Exponential decaying | $$ a^n \cdot u[n] $$ | $ a $ real or complex with $ |a| < 1 $ |
| Complex | $$ e^{jn\omega_0} $$ | $$ \cos(n\omega_0) + j\sin(n\omega_0)$$ |
Signal duration
Signal properties
\begin{eqnarray*} \textit{Periodic} &:& x[n]=x[n+N] \newline \textit{Even } \color{blue}{\textit{Conjugate-symmetric}} &:& x[n]=x[-n] \hspace{3mm} \color{blue}{x[n]=x^\ast[-n]}\newline \textit{Odd } \color{blue}{\textit{Conjugate-antisymmetric}} &:& x[n]=-x[-n] \hspace{3mm} \color{blue}{x[n]=-x^\ast[-n]} \newline \textit{General: } \hspace{3mm} x[n]&=&x_e[n]+x_o[n] \newline \text{with } \hspace{3mm} x_e[n]&=&\frac{1}{2} \left ( x[n] + x[-n] \right ) \newline \text{and } \hspace{3mm} x_o[n]&=&\frac{1}{2} \left ( x[n] - x[-n] \right ) \end{eqnarray*}
Signal manipulations
Amplitude transformations: \begin{eqnarray*} \color{blue}{\textit{Addition}} &:& y[n]=x_1[n]+x_2[n] \newline \color{blue}{\textit{Multiplication}} &:& y[n]=x_1[n] \cdot x_2[n] \newline \color{blue}{\textit{Scaling}} &:& y[n] = c \cdot x[n] \end{eqnarray*}
Transformation of independent variable $n$:
\begin{eqnarray*} \textit{Time shifting (delay or advance)} &:& f[n]=n-n_0 \newline \textit{Time reversal} &:& f[n]=-n \newline \textit{Time scaling = Down- or Up- sampling} &:& f[n]=M \cdot n \text{ or } f[n]=n/M \end{eqnarray*} Note: Shifting, reversal, and time scaling operations are order dependent.