# OpenStax-CNX

You are here: Home » Content » Information and Signal Theory » Discrete time signals

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Anders Gjendemsjø. E-mail the author

# Discrete time signals

Module by: Anders Gjendemsjø. E-mail the author

Summary: Important discrete time signals

The signals and relations presented in this module are quite similar to those in the Analog signals module. So do compare and find similarities and differences!

## Sequences

Generally a time discrete signal is a sequence of real or complex numbers. Each component in the sequence is identified by an index: ...x(n-1),x(n), x(n+1),...

### Example 1

[x(n)] = [0.5 2.4 3.2 4.5] is a sequence. Using the index to identify a component we have x0=0.5 x 0 0.5 , x1=2.4 x 1 2.4 and so on.

## Manipulating sequences

• Multiplication by a constant: Multiply every component by the constant
• Multiplication of sequences: Multiply each component individually
• Delay: A delay by kk implies that we shift the sequence by k. For this to make sense the sequence has to be of infinite length.

### Example 2

Given the sequences [x(n)] = [0.5 2.4 3.2 4.5] and [y(n)] = [0.0 2.2 7.2 5.5].

b)Multiplication by a constant c=2. [w(n)]= 2 *[x(n)] = [1.0 4.8 6.4 9.0]

## Elementary signals & relations

### The unit sample

The unit sample is a signal which is zero everywhere except when its argument is zero, then it is equal to 1. Mathematically

#### unit sample:

δn={1  if  n=00  otherwise   δ n 1 n0 0
The unit sample function is very useful in that it can be seen as the elementary constituent in any discrete signal. Let xnxn be a sequence. Then we can express xnxn as follows (using the unit sample definition and the delay operation)
xn=k=xkδnk x n k x k δ n k
(1)

### The unit step

The unit step function is equal to zero when its index is negative and equal to one for non-negative indexes, see Figure 1 for plots.

#### unit step:

un={1  if  n00  otherwise   u n 1 n0 0

### Trigonometric functions

The discrete trigonometric functions are defined as follows. nn is the sequence index and ωω is the angular frequency. ω=2πf ω 2 f , where f is the digital frequency.

xn=sinωn x n ω n

xn=cosωn x n ω n

### The complex exponential function

The complex exponential function is central to signal processing and some call it the most important signal. Remember that it is a sequence and that j=1 1 is the imaginary unit.

xn=ejωn x n ω n

## Euler's relations

The complex exponential function can be written as a sum of its real and imaginary part.

xn=ejωn=cosωn+jsinωn x n ω n ω n ω n
(2)
By complex conjugating Equation 2 and add / subtract the result with Equation 2 we obtain Euler's relations.

### Euler relation 1:

cosωn=ejωn+e(jωn)2 ω n ω n ω n 2

### Euler relation 2:

sinωn=ejωne(jωn)2j ω n ω n ω n 2
The importance of Euler's relations can hardly be stressed enough.

Take a look at ?

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

#### Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

#### Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks