Chapter 1: Elementary Discrete Signals

Introduction

This chapter presents the fundamental discrete-time signals that form the building blocks of digital signal processing.

Unit Impulse

Definition: Unit Impulse (Kronecker Delta)

The discrete unit impulse is defined as:

$$\delta [n]=\{\begin{array}{ll}1& \text{if }n=0\\ 0& \text{if }n\ne 0\end{array}$$

Sifting Property

For any signal $x[n]$:

$$\sum _{k=-\mathrm{\infty }}^{+\mathrm{\infty }}x[k]\delta [n-k]=x[n]$$$$x[n]\delta [n-n_0]=x[n_0]\delta [n-n_0]$$

Unit Step

Definition: Unit Step

The discrete unit step is defined as:

$$u[n]=\{\begin{array}{ll}1& \text{if }n\ge 0\\ 0& \text{if }n<0\end{array}$$

Relationship with Unit Impulse

$$\delta [n]=u[n]-u[n-1]$$$$u[n]=\sum _{k=-\mathrm{\infty }}^n\delta [k]=\sum _{k=0}^{\mathrm{\infty }}\delta [n-k]$$

Exponential Sequences

Definition: Real Exponential

A real exponential sequence is:

$$x[n]=A{\alpha }^n$$

where $A$ is the amplitude and $\alpha$ is the base.

• $|\alpha |<1$: decaying sequence
• $|\alpha |>1$: growing sequence
• $|\alpha |=1$: constant magnitude

Definition: Complex Exponential

The discrete complex exponential is:

$$x[n]=A{e}^{j({\omega }_{0}n+\varphi )}$$

where ${\omega }_0$ is the normalized angular frequency (rad/sample).

Periodicity of Complex Exponentials

Theorem: Periodicity Condition

The discrete complex exponential $e^{j{\omega }_{0}n}$ is periodic if and only if $\frac{{\omega }_0}{2\pi }$ is a rational number.

If $\frac{{\omega }_0}{2\pi }=\frac{m}{N}$ where $m$ and $N$ are coprime integers, the fundamental period is $N$.

Example: Periodicity

Consider $x[n]=e^{j(2\pi /12)n}$.

Since $\frac{{\omega }_0}{2\pi }=\frac{1}{12}$ is rational, the signal is periodic with period $N=12$.

Compare with $x[n]=e^{j\sqrt{2}n}$: since $\frac{\sqrt{2}}{2\pi }$ is irrational, this signal is not periodic.

Sinusoidal Sequences

A discrete sinusoid is:

$$x[n]=A\cos ({\omega }_{0}n+\varphi )$$

where:

• ${\omega }_0$: normalized angular frequency (rad/sample)
• ${\omega }_0=2\pi f_0/f_s=2\pi f_0{T}_s$: relationship with analog frequency

Proposition: Frequency Aliasing

Discrete sinusoids with frequencies ${\omega }_0$ and ${\omega }_0+2\pi k$ (for any integer $k$) are identical:

$$\cos (({\omega }_0+2\pi k)n+\varphi )=\cos ({\omega }_{0}n+\varphi )$$

Therefore, the unique frequency range is $-\pi <{\omega }_0\le \pi$ or $0\le {\omega }_0<2\pi$.

Geometric Sequence

Definition: Geometric Sequence

A causal geometric sequence is:

$$x[n]=a^{n}u[n]$$

The sum of an infinite geometric series (for $|a|<1$):

$$\sum _{n=0}^{\mathrm{\infty }}{a}^n=\frac{1}{1-a}$$

Rectangular Window

Definition: Rectangular Window

The rectangular window of length $N$ is:

$$w[n]=\{\begin{array}{ll}1& 0\le n\le N-1\\ 0& \text{otherwise}\end{array}$$

Or equivalently: $w[n]=u[n]-u[n-N]$

Digital Sinc Function

Definition: Discrete Sinc

The discrete sinc function is:

$${\text{sinc}}_d[n]=\frac{\sin (\pi n)}{\pi n}=\{\begin{array}{ll}1& n=0\\ 0& n\ne 0,n\in \mathbb{Z}\end{array}$$

The Dirichlet sinc (aliased sinc) is:

$$\frac{\sin ({\omega }_{c}n)}{\pi n}$$

which is the ideal lowpass filter impulse response with cutoff ${\omega }_c$.