Skip to content Skip to navigation


You are here: Home » Content » Properties of LTI Systems


Recently Viewed

This feature requires Javascript to be enabled.

Properties of LTI Systems

Module by: Richard Baraniuk. E-mail the author

Summary: Introduction to the properties of LTI systems.

All LTI systems are characterized by the convolution Figure 1

yn= k =xkhnk y n k x k h n k
Figure 1
Figure 1 (fig10.png)
  • The impulse response hn h n is a complete characterization of the properties of a specific LTI system.
  • For LTI systems, we often use this diagram.
Figure 2
Figure 2 (fig9.png)
Let's look at some of these properties and how they are reflected in hn h n .

Convolution is Commutative

This means:

xn*hn=hn*xn x n h n h n x n
or Figure 3.
Figure 3
Figure 3 (fig8.png)


The proof is a simple change of variables:

yn=xn*hn= k =xkhnk= m =xnmhm= m =hmxnm=hn*xn y n x n h n k x k h n k m x n m h m m h m x n m h n x n
where m=nk m n k , which means k=nm k n m .

Cascade Connecion of LTI Systems

Compute Figure 4.

Figure 4
Figure 4 (fig7.png)

Two Methods

  1. Brute Force and Massive Ignorance: Use yn=vn* h 2 n y n v n h 2 n and substitute in vn=xn* h 1 n v n x n h 1 n . Try this at home.
  2. Elegant and Quick: Just find the overall impulse response, Figure 5, which implies Figure 6.
Figure 5: h 1 n* h 2 n h 1 n h 2 n is the overal impulse response.
Figure 5 (fig6.png)
Figure 6
Figure 6 (fig5.png)

Parallel Connection of LTI Systems

Same elegant reasoning gives Figure 7.

Figure 7
Figure 7 (fig4.png)

Example 1: Moving Window Filter

Figure 8
Figure 8 (fig3.png)

What is the impulse response of Figure 8?

Is it an FIR or IIR? Figure 8?

What is yn y n of Figure 8?

Example 2: Pure Recursive Filter

Figure 9
Figure 9 (fig2.png)

What is the impulse response of Figure 9?

Is it an FIR or IIR ? Figure 9?

What is yn y n of Figure 9?

Proof That the Filter Structure Computes the Correct Output

Find its impulse response (Figure 10) and show that it is hh.

Table 1
nn Input δ 0 n δ 0 n
0 1 0 0 0 0 0
1 0 1 0 0 0 0
2 0 0 1 0 0 0
3 0 0 0 1 0 0
4 0 0 0 0 1 0
Figure 10
Figure 10 (fig1.png)
Table 2
n n hn h n
0 h0 h 0
1 h1 h 1
2 h2 h 2
3 h3 h 3
4 h4 h 4
where the hn h n 's make up h h, which is the impulse response we desire.

Example 3: Combination Moving Window/Recursive Filter

Figure 11.

Figure 11
Figure 11 (figlast.png)

Content actions

Download module as:

PDF | EPUB (?)

What is an EPUB file?

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

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add 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


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? tag icon

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