Discrete-time systems

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:

1. System properties - Systems can be characterized in multiple ways. These properties will be discussed in this section.
2. System visualization - This section will introduce the basic building blocks of a discrete-time systems.
3. 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.
4. Convolution - The operation performed by the filter can be denoted mathematically using the convolution operation.
5. 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.
6. 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

$$x\left[n\right]\mapsto y\left[n\right]\text{;}{x}_1\left[n\right]\mapsto y_1\left[n\right]\text{;}{x}_2\left[n\right]\mapsto y_2\left[n\right]$$

$$\begin{array}{ccc}\text{Memoryless}& :& \text{Output at}n=n_0\text{depends only on input at}n=n_0\\ \text{Causality}& :& \text{Response at}{n}_0\text{depends on input up to}n=n_0\\ \text{Invertibility}& :& \text{Input may be}\text{uniquely}\text{determined from output}\\ \text{Additivity}& :& x\left[n\right]=x_1\left[n\right]+x_2\left[n\right]\mapsto y\left[n\right]=y_1\left[n\right]+y_2\left[n\right]\\ \text{Homogeneity}& :& c\cdot x\left[n\right]\mapsto c\cdot y\left[n\right]\\ \text{Impulse response}& :& \text{Response to}\delta \left[n\right]\Rightarrow \delta \left[n\right]\mapsto h\left[n\right]\\ \text{(BIBO) Stability}& :& \text{For}A,B<\infty \text{,}|x\left[n\right]|<A\Rightarrow |y\left[n\right]|<B\stackrel{\text{LTI}}{\iff }\sum _{n=-\infty }^{\infty }|h\left[n\right]|<\infty \\ \text{Linearity}& :& x\left[n\right]=\alpha x_1\left[n\right]+\beta x_2\left[n\right]\mapsto y\left[n\right]=\alpha y_1\left[n\right]+\beta y_2\left[n\right]\\ \text{Time-Invariance}& :& x[n-n_0\mapsto y[n-n_0]\\ \text{LTI}& :& \text{Linear Time-Invariance}\end{array}$$

Basic building blocks

Realization scheme

Difference Equation

$$y\left[n\right]=\sum _{k=0}^{M-1}{b}_{k}x[n-k]+\sum _{k=1}^{N-1}{a}_{k}y[n-k]$$

Impulse response properties LTI

$$\begin{array}{ccc}\text{BIBO stablility}& :& \sum _{n=-\infty }^{\infty }|h\left[n\right]|<\infty \\ \text{Causalilty}& :& h\left[n\right]=0\text{for}n<0\end{array}$$

Convolution sum

$$y\left[n\right]=\sum _{k=-\infty }^{\infty }x\left[k\right]h[n-k]=x\left[n\right]\star h\left[n\right]$$

Convolution sum procedure:

1. Plot both input $x$ and impulse response $h$ as function of index $k$
2. Mirror (reverse) $h\left[k\right]$ $\Rightarrow$ $h[-k]$
3. For each new index $n$:
   1. Shift mirrored $h[-k]$ to index $n$ $\Rightarrow$ $h[n-k]$.
   2. Output $y\left[n\right]$ is equal to the result of the sum of element by element multiplications of $x\left[k\right]$ and $h[n-k]$.

When the length of an input sequence of signal samples is $N$ and the length of the sequence of impulse response values is $M$, then the convolution sum procedure results in a sequence of length $N+M-1$ output signal samples $y\left[n\right]$.

Convolution sum properties

$$\begin{array}{ccc}\text{Commutative}& :& h\left[n\right]\star x\left[n\right]=x\left[n\right]\star h\left[n\right]\\ \text{Associative}& :& \left\{x\right[n]\ast h_1[n\left]\right\}\ast h_2\left[n\right]=x\left[n\right]\ast \left\{h_1\right[n]\ast h_2[n\left]\right\}\\ \text{Distributive}& :& x\left[n\right]\ast h_1\left[n\right]+x\left[n\right]\ast h_2\left[n\right]=x\left[n\right]\ast \left\{h_1\right[n]+h_2[n\left]\right\}\end{array}$$

Cascade equivalence properties LTI system

• The cascading of two LTI systems, one with impulse response $h_1\left[n\right]$ and the other one with impulse response $h_2\left[n\right]$, can be combined to one LTI system with impulse response $h\left[n\right]=h_1\left[n\right]\star h_2\left[n\right]$.
• The order of two LTI systems, one with impulse response $h_1\left[n\right]$ and the other one with $h_2\left[n\right]$, can be interchanged.