Introduction
This module introduces the concept of processing a discrete-time signal by a discrete-time system and the classification of such systems.
Screencast video [⯈]
Module overview
This module will cover the following topics:
- System properties - Systems can be characterized in multiple ways. These properties will be discussed in this section.
- System visualization - This section will introduce the basic building blocks of a discrete-time systems.
- Impulse response - A system can be characterized by its impulse response, which represents the output of the first when a short pulse is applied to its input.
- Convolution - The operation performed by the filter can be denoted mathematically using the convolution operation.
- Cascading systems - When a signal is passed through multiple systems, these signals may be combined, such that the signal is passing to another single filter.
- Special convolutions [⯈] - When dealing with real-time signals or when having limited memory, the convolution operation might not be ideal, therefore some special convolution operations are introduced that solve these problems.
Summary
Most important system properties
$$ x[n] \mapsto y[n] \text{ ; } x_1[n] \mapsto y_1[n] \text{ ; } x_2[n] \mapsto y_2[n] $$
\begin{eqnarray*} \color{blue}{\textit{Memoryless}} &:& \text{Output at $n=n_0$ depends only on input at $n=n_0$}\newline \color{blue}{\textit{Causality}} &:& \text{Response at $n_0$ depends on input up to $n=n_0$}\newline \color{blue}{\textit{Invertibility}} &:& \text{Input may be }\color{red}{\text{uniquely}}\text{ determined from output}\newline \color{blue}{\textit{Additivity}} &:& x[n]=x_1[n]+x_2[n] \mapsto y[n]=y_1[n]+y_2[n]\newline \color{blue}{\textit{Homogeneity}} &:& c \cdot x[n] \mapsto c \cdot y[n]\newline \color{blue}{\textit{Impulse response}} &:& \text{Response to } \delta[n] \hspace{2mm} \Rightarrow \delta[n] \mapsto h[n] \newline \color{blue}{\textit{(BIBO) Stability}} &:& \text{For } A,B < \infty \text{,} \hspace{3mm} |x[n]| < A \hspace{2mm} \Rightarrow \hspace{2mm} |y[n]| < B \text{ } \overset{\color{red}{\text{LTI}}}{\Leftrightarrow} \text{ } \sum_{n=-\infty}^{\infty} |h[n]| < \infty \newline \color{blue}{\textit{Linearity}} &:& x[n]= \alpha x_1[n] + \beta x_2[n] \mapsto y[n] = \alpha y_1[n] + \beta y_2[n] \newline \color{blue}{\textit{Time-Invariance}} &:& x[n-n_0 \mapsto y[n-n_0]\newline \color{red}{\textit{LTI}} &:& \color{red}{\text{Linear Time-Invariance}} \end{eqnarray*}
Basic building blocks
Realization scheme
Difference Equation
$$ y[n] = \sum_{k=0}^{M-1} b_k x[n-k] + \sum_{k=1}^{N-1} a_k y[n-k] $$
Impulse response properties LTI
\begin{eqnarray*} \textbf{BIBO stablility} & : & \sum_{n=-\infty}^{\infty} |h[n]| < \infty \newline \textbf{Causalilty} & : & h[n]=0 \text{ for } n<0 \end{eqnarray*}
Convolution sum
$$ y[n] = \sum_{k=-\infty}^{\infty} x[k] h[n-k]= x[n] \star h[n] $$
Convolution sum procedure:
- Plot both input $x$ and impulse response $h$ as function of index $k$
- Mirror (reverse) $h[k]$ $\Rightarrow$ $h[-k]$
- For each new index $n$:
- Shift mirrored $h[-k]$ to index $n$ $\Rightarrow$ $h[n-k]$.
- Output $y[n]$ is equal to the result of the sum of element by element multiplications of $x[k]$ and $h[n-k]$.
Convolution sum properties
\begin{eqnarray*} \text{Commutative} &:& h[n] \star x[n] = x[n] \star h[n] \newline \text{Associative} &:& \{ x[n] \ast h_1[n] \} \ast h_2[n] = x[n] \ast \{ h_1[n] \ast h_2[n] \} \newline \text{Distributive} &:& x[n] \ast h_1[n] + x[n] \ast h_2[n] = x[n] \ast \{ h_1[n] + h_2[n] \} \end{eqnarray*}
Cascade equivalence properties LTI system
- The cascading of two LTI systems, one with impulse response $h_1[n]$ and the other one with impulse response $h_2[n]$, can be combined to one LTI system with impulse response $h[n]=h_1[n] \star h_2[n]$.
- The order of two LTI systems, one with impulse response $h_1[n]$ and the other one with $h_2[n]$, can be interchanged.