Connexions

You are here: Home » Content » Properties of Systems

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.

Properties of Systems

Module by: Thanos Antoulas, JP Slavinsky. E-mail the authors

Summary: Properties of different types of systems

Note: You are viewing an old version of this document. The latest version is available here.

Linear Systems

If a system is linear, this means that when an input to a given system is scaled by a value, the output of the system is scaled by the same amount.

In Figure 1(a) above, an input xx to the linear system LL gives the output yy. If xx is scaled by a value αα and passed through this same system, as in Figure 1(b), the output will also be scaled by αα.

A linear system also obeys the principle of superposition. This means that if two inputs are added together and passed through a linear system, the output will be the sum of the individual inputs' outputs.

That is, if Figure 2 is true, then Figure 3 is also true for a linear system. The scaling property mentioned above still holds in conjunction with the superposition principle. Therefore, if the inputs x and y are scaled by factors α and β, respectively, then the sum of these scaled inputs will give the sum of the individual scaled outputs:

Time-Invariant Systems

A time-invariant system has the property that a certain input will always give the same output, without regard to when the input was applied to the system.

In this figure, xtxt and xtt0 xtt0 are passed through the system TI. Because the system TI is time-invariant, the inputs xtxt and xtt0 xtt0 produce the same output. The only difference is that the output due to xtt0 xtt0 is shifted by a time t0t0.

Whether a system is time-invariant or time-varying can be seen in the differential equation (or difference equation) describing it. Time-invariant systems are modeled with constant coefficient equations. A constant coefficient differential (or difference) equation means that the parameters of the system are not changing over time and an input now will give the same result as the same input later.

Linear Time-Invariant (LTI) Systems

Certain systems are both linear and time-invariant, and are thus referred to as LTI systems.

As LTI systems are a subset of linear systems, they obey the principle of superposition. In the figure below, we see the effect of applying time-invariance to the superposition definition in the linear systems section above.

LTI Systems in Series

If two or more LTI systems are in series with each other, their order can be interchanged without affecting the overall output of the system. Systems in series are also called cascaded systems.

LTI Systems in Parallel

If two or more LTI systems are in parallel with one another, an equivalent system is one that is defined as the sum of these individual systems.

Causality

A system is causal if it does not depend on future values of the input to determine the output. This means that if the first input to a system comes at time t0t0, then the system should not give any output until that time. An example of a non-causal system would be one that "sensed" an input coming and gave an output before the input arrived:

A causal system is also characterized by an impulse response htht that is zero for t<0t0.

Content actions

Give feedback:

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

Reuse / Edit:

Reuse or edit module (?)

Check out and edit

If you have permission to edit this content, using the "Reuse / Edit" action will allow you to check the content out into your Personal Workspace or a shared Workgroup and then make your edits.

Derive a copy

If you don't have permission to edit the content, you can still use "Reuse / Edit" to adapt the content by creating a derived copy of it and then editing and publishing the copy.