Chapter 2: Discrete LTI Systems

Introduction

Discrete-time Linear Time-Invariant systems are fundamental to digital signal processing, with properties analogous to their continuous-time counterparts.

System Properties

Definition: Discrete-Time System

A discrete-time system transforms an input sequence $x[n]$ into an output sequence $y[n]$:

$$y[n]=\mathcal{T}\{x[n]\}$$

Linearity

A system is linear if:

$$\mathcal{T}\{a_1{x}_1[n]+a_2{x}_2[n]\}=a_1\mathcal{T}\{x_1[n]\}+a_2\mathcal{T}\{x_2[n]\}$$

Time-Invariance

A system is time-invariant if:

$$\text{If }y[n]=\mathcal{T}\{x[n]\},\text{ then }y[n-n_0]=\mathcal{T}\{x[n-n_0]\}$$

Impulse Response

Definition: Impulse Response

The impulse response $h[n]$ of a discrete LTI system is the output when the input is a unit impulse:

$$h[n]=\mathcal{T}\{\delta [n]\}$$

Discrete Convolution

Definition: Convolution Sum

The output of an LTI system is the convolution of input and impulse response:

$$y[n]=x[n]\ast h[n]=\sum _{k=-\mathrm{\infty }}^{+\mathrm{\infty }}x[k]h[n-k]$$

Properties of Discrete Convolution

Proposition: Convolution Properties

1. Commutative: $x[n]\ast h[n]=h[n]\ast x[n]$

2. Associative: $(x[n]\ast h_1[n])\ast h_2[n]=x[n]\ast (h_1[n]\ast h_2[n])$

3. Distributive: $x[n]\ast (h_1[n]+h_2[n])=x[n]\ast h_1[n]+x[n]\ast h_2[n]$

4. Identity: $x[n]\ast \delta [n]=x[n]$

5. Shift: $x[n]\ast \delta [n-n_0]=x[n-n_0]$

Example: Convolution of Finite Sequences

Let $x[n]=\{1,2,3\}$ for $n=0,1,2$ and $h[n]=\{1,1\}$ for $n=0,1$.

$$y[n]=x[n]\ast h[n]=\{1,3,5,3\}$$

for $n=0,1,2,3$. Note: if $x[n]$ has length $L$ and $h[n]$ has length $M$, then $y[n]$ has length $L+M-1$.

Causality and Stability

Causality

Definition: Causal System

A discrete LTI system is causal if:

$$h[n]=0\text{for }n<0$$

BIBO Stability

Theorem: Stability Criterion

A discrete LTI system is BIBO stable if and only if:

$$\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}|h[n]|<\mathrm{\infty }$$

Difference Equations

Definition: Linear Constant-Coefficient Difference Equation

Many discrete LTI systems are described by:

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

Or in recursive form:

$$y[n]=\frac{1}{a_0}(\sum _{k=0}^M{b}_{k}x[n-k]-\sum _{k=1}^N{a}_{k}y[n-k])$$

FIR and IIR Systems

Definition: FIR System

A Finite Impulse Response (FIR) system has an impulse response of finite duration:

$$h[n]=0\text{for }n<0\text{ and }n\ge M$$

FIR systems are always stable and are non-recursive (no feedback).

Definition: IIR System

An Infinite Impulse Response (IIR) system has an impulse response of infinite duration. IIR systems are recursive and require stability analysis.

Example: Moving Average Filter (FIR)

A 3-point moving average filter:

$$y[n]=\frac{1}{3}(x[n]+x[n-1]+x[n-2])$$

Impulse response: $h[n]=\frac{1}{3}$ for $n=0,1,2$, and $h[n]=0$ otherwise.

Example: First-Order Recursive Filter (IIR)

$$y[n]=x[n]+ay[n-1]$$

Impulse response: $h[n]=a^{n}u[n]$

Stable if $|a|<1$.

Frequency Response

Definition: Frequency Response

The frequency response of a discrete LTI system is:

$$H(e^{j\omega })=\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}h[n]e^{-j\omega n}$$

This is the DTFT of the impulse response.

Theorem: Eigenfunction Property

Complex exponentials are eigenfunctions of discrete LTI systems:

$$\mathcal{T}\{e^{j\omega n}\}=H(e^{j\omega })e^{j\omega n}$$