Skip to content Skip to navigation

Connexions

You are here: Home » Content » BIBO Stability

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

BIBO Stability

Module by: Richard Baraniuk. E-mail the author

Summary: An overview of BIBO stability.

With finite length signals ∈ k k N , it is impossible to make the system output blow up.

But this can happen with ∞ -length system.

Example 1

Figure 1
Figure 1 (fig1.png)
y⁢n=∑ k =−∞∞h⁢n−k⁢x⁢k=∑ k =−∞∞1×1=∞ y n k h n k x k k 1 1 for all n n

l p ⁢Z l p spaces are useful to understand what's going on...

Definition 1: Bounded-Input Bounded-Output (BIBO) Stable
When, in an LSI system, for any bounded input the output is always bounded.
Figure 2
Figure 2 (fig2.png)
y⁢n=∑ k =−∞∞h⁢n−k⁢x⁢k y n k h n k x k −∞<n<∞ n
Figure 3
Figure 3 (fig3.png)
bounded input/output →
x, y∈ l ∞ ⁢Z x, y l ∞
(1)
So, the question is what is the condition on h h such that (y=h*x)∈ l ∞ ⁢Z y h x l ∞ with x∈ l ∞ ⁢Z x l ∞ ?

Fact:

An LSI system is BIBO stable if and only if the impulse response h∈ l 1 ⁢Z h l 1 .
That is, if x∈ l ∞ ⁢∞ x l ∞ and h∈ l 1 ⁢Z h l 1 then (y=h*x)∈ l ∞ ⁢Z y h x l ∞

Proof

Not that the fact is true "if and only if" → there are two sides to the proof.

Part 1

Sufficiency: If h∈ l 1 ⁢Z h l 1 then system is BIBO. ∥y∥∞=max n∈Z n∈Z |y⁢n| y n y n So look at |y⁢n| y n and show it is <∞ .

Use convolution formula |y⁢n|=|∑ k =−∞∞x⁢n−k⁢h⁢k| y n k x n k h k ≤∑ k =−∞∞|x⁢n−k|⁢|h⁢k| k x n k h k Where x∈ l ∞ ⁢Z→(|x⁢n−k|≤∥x∥∞) x l ∞ x n k x ≤∑ k =−∞∞|h⁢k|⁢∥x∥∞ k h k x Since ∥x∥∞ x is constant, =∥x∥∞⁢∑ k =−∞∞|h⁢k| x k h k Where ∑ k =−∞∞|h⁢k|=∥h∥1 k h k 1 h Therefore, |y⁢n|≤∥x∥∞⁢∥h∥1 y n x 1 h But x∈ l ∞ ⁢Z→(∥x∥∞<∞) x l ∞ x h∈ l 1 ⁢Z→(∥h∥1<∞) h l 1 1 h Therefore |y⁢n|<∞ y n Therefore max n∈Z n∈Z |y⁢n|<∞ n y n Therefore ∥y∥∞<∞ y and the bounded input x x created a bounded output y y.

Part 2:

Necessity: If h∉ l 1 ⁢Z h l 1 then the system is not BIBO.

Note:

This part is tricky!!!
So, assume that ∥h∥1=∞ 1 h and show that there exists a signal x∈ l ∞ ⁢Z x l ∞ such that y∈ l ∞ ⁢Z y l ∞ .

Note:

We only have 1 x x to show this.
Some creativity suggests the following signal: x⁢n={h⁢−n¯|h⁢−n|  if  h⁢n≠0 0   if  h⁢=0 x n h n h n h n 0 0 h 0 For real h h, x x is the sign of time-reversed h h.
Figure 4
Figure 4 (fig4.png)

Note:

x x takes on values -1, 1, 0, and so ∥x∥∞=1<∞ x 1 ie: bounded input.
Compute y⁢n y n at n=0 n 0 : y⁢n=∑ k =−∞∞x⁢n−k⁢h⁢k y n k x n k h k y⁢0=∑ k =−∞∞x⁢−k⁢h⁢k y 0 k x k h k x⁢−k=h⁢−k¯|h⁢−k| x k h k h k =∑ k =−∞∞h⁢k¯⁢h⁢k|h⁢k| k h k h k h k =∑ k |h⁢k|2|h⁢k| k h k 2 h k =∑ k =−∞∞|h⁢k| k h k y⁢0=∥h∥1=∞ y 0 1 h by assumption.

→y∉ l ∞ ⁢Z y l ∞ This means that it is not bounded → system is not BIBO.

BIBO Stablility Bottom Line

Figure 5
Figure 5 (fig5.png)
x∈ l ∞ ⁢Z x l ∞ bounded and y∈ l ∞ ⁢Z y l ∞ bounded.

Need: h∈ l 1 ⁢Z h l 1 ie: ∥h∥1<∞ 1 h and h h must decay fast enough as |n|→∞ n .

Example 2: Moving Window Filter (FIR)

Is it BIBO stable?

Example 3: Recursive Filter (IIR)

Is it BIBO stable?

Content actions

Download module as:

PDF | 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

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? 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