# Connexions

You are here: Home » Content » DSPA » Systems in the Time-Domain

### Lenses

What is a lens?

#### 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.

#### In these lenses

• SigProc

This module is included inLens: Signal Processing
By: Daniel McKennaAs a part of collection: "Fundamentals of Signal Processing"

Click the "SigProc" link to see all content selected in this lens.

Click the tag icon to display tags associated with this content.

### Recently Viewed

This feature requires Javascript to be enabled.

### Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Inside Collection (Course):

Course by: Janko Calic. E-mail the author

# Systems in the Time-Domain

Module by: Don Johnson. E-mail the author

Summary: Discrete-time systems allow for mathematically specified processes like the difference equation.

A discrete-time signal sn sn is delayed by n0 n0 samples when we write snn0 s n n0 , with n0>0 n0 0 . Choosing n0 n0 to be negative advances the signal along the integers. As opposed to analog delays, discrete-time delays can only be integer valued. In the frequency domain, delaying a signal corresponds to a linear phase shift of the signal's discrete-time Fourier transform: sn n 0 e(j2πf n 0 )Sej2πf s n n 0 2 f n 0 S 2 f .

Linear discrete-time systems have the superposition property.

## Superposition

S a 1 x 1 n+ a 2 x 2 n= a 1 S x 1 n+ a 2 S x 2 n S a 1 x 1 n a 2 x 2 n a 1 S x 1 n a 2 S x 2 n
(1)
A discrete-time system is called shift-invariant (analogous to time-invariant analog systems) if delaying the input delays the corresponding output.

## Shift-Invariant

If   Sxn=yn , Then   Sxn n 0 =yn n 0 If   S x n y n , Then   S x n n 0 y n n 0
(2)
We use the term shift-invariant to emphasize that delays can only have integer values in discrete-time, while in analog signals, delays can be arbitrarily valued.

We want to concentrate on systems that are both linear and shift-invariant. It will be these that allow us the full power of frequency-domain analysis and implementations. Because we have no physical constraints in "constructing" such systems, we need only a mathematical specification. In analog systems, the differential equation specifies the input-output relationship in the time-domain. The corresponding discrete-time specification is the difference equation.

## The Difference Equation

yn= a 1 yn1++ a p ynp+ b 0 xn+ b 1 xn1++ b q xnq y n a 1 y n 1 a p y n p b 0 x n b 1 x n 1 b q x n q
(3)
Here, the output signal yn yn is related to its past values ynl y n l , l=1p l 1 p , and to the current and past values of the input signal xn xn. The system's characteristics are determined by the choices for the number of coefficients pp and qq and the coefficients' values a1ap a1 ap and b0b1bq b0 b1 bq .

## Aside:

There is an asymmetry in the coefficients: where is a 0 a 0 ? This coefficient would multiply the yn y n term in the difference equation. We have essentially divided the equation by it, which does not change the input-output relationship. We have thus created the convention that a0a0 is always one.

As opposed to differential equations, which only provide an implicit description of a system (we must somehow solve the differential equation), difference equations provide an explicit way of computing the output for any input. We simply express the difference equation by a program that calculates each output from the previous output values, and the current and previous inputs.

## 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.

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