Connexions

You are here: Home » Content » Size of A Signal: Norms
Content Actions
Lenses

What is a lens?

Lenses

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

What is in a lens?

Lens makers point to Connexions 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 Connexions member, a community, or a respected organization.

This content is ...
Affiliated with (?)
This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.

  • This module is included inLens: Rice University OpenCourseWare
    By: OpenCourseWare ConsortiumAs a part of collection:"Signals and Systems"

    Click the "Rice University OCW" link to see all content affiliated with them.

    Rice University OCW
Also in these lenses

  • This module is included inLens: richb's DSP resources
    By: Richard BaraniukAs a part of collection:"Signals and Systems"

    Comments:

    "My introduction to signal processing course at Rice University."

    Click the "richb's DSP" link to see all content selected in this lens.

    richb's DSP
Tags

(?)

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

Size of A Signal: Norms

Module by: Richard Baraniuk

Summary: A module concerning the size of a signal, more specifically norms.

"Size" indicates largeness or strength. We will use the mathematical concept of the norm to quantify this notion for both continuous-time and discrete-time signals. First we consider a way to quantify the size of a signal which may already be familiar.

Continuous-Time Energy

Our usual notion of the energy of a signal is the area under the curve |ft|2 f t 2
figure1.png
Figure 1
E f =∫-∞∞|ft|2dt E f t f t 2 (1)
Example 1 
Calculate E f E f for
figure2.png
Figure 2

Generalized Energy: Norms

The notion of "energy" can be generalized through the introduction of the L p L p norm:
∥f∥p=∫|ft|pdt1p p f t f t p 1 p (2)
where 1≤p<∞ 1 p .
Example 2 
E f =∥f∥22 E f 2 f 2
Example 3 
Calculate the L p L p norm of
figure3a.png
Figure 3
Problem 1
What happens to ∥f∥p=∫|ft|pdt1p p f t f t p 1 p as p→∞ p ?
L ∞ L ∞ norm ∥f∥∞=ess sup|ft| ∞ f ess sup f t
figure3.png
Figure 4

Discrete-Time Energy

figure4.png
Figure 5
E f =∑n=-∞∞|fn|2 E f n f n 2
∥f∥p=∫|ft|pdt1p p f t f t p 1 p where 1≤p<∞ 1 p
∥f∥∞=maxn{|fn|} f n f n ∥f∥pp=∑n=0N-1|fn|p p f p n 0 N 1 f n p where 1≤p<∞ 1 p ∥f∥∞=maxn=0, 1, ..., N-1{|fn|} f n 0, 1, ..., N-1 f n

Finite Norm Signals

What are the conditions on a signal for ∥f∥p<∞ p f ? Look at all 4 fundamental signal classes

Discrete-Time and Finite Length

figure5.png
Figure 6
This is a length N N vector. f=f0f1f2f...fN-1= f 0 f 1 f 2 ... f N 1 f f 0 f 1 f 2 f ... f N 1 f f 0 f f 1 f f 2 ... f f N 1 where f∈ℂN f N , or f∈N f N N N-dimensional complex or real Euclidean space.
Example 4 
N=3 N 3 , f f is a real signal.
figure6a.png
Figure 7
Definition 1:
l p 0N-1= f∈ℂN∥f∥p<∞ l p 0 N 1 f N p f but from previous discussion l p 0N-1=ℂN l p 0 N 1 N

Discrete-Time and Infinite Length

figure6.png
Figure 8
can still interpret f f as an infinite-length vector f=f...f-1f0f1f2f... f f ... f -1 f 0 f 1 f 2 f ... but ℂ∞ , R∞ R don't make sense.
Definition 2:
l p z= f∥f∥p<∞ l p z f p f ∥f∥pp=∑n=-∞∞|fn|p p f p n f n p where 1≤p<∞ 1 p ∥f∥∞=maxn∈z{|fn|} f n z f n
What does it take for an f f to be in l p z l p z ?
Example 5 
Sketch an f∈ l p z f l p z and f∉ l p z f l p z .
Problem 2
What characteristics does f∈ l p z f l p z have and what happens as we charge p p?

Continuous-Time and Finite-Length

figure7.png
Figure 9
We will still refer to ft f t as a vector; more on this later.
Definition 3:
L p T 1 T 2 = f T 1 T 2 ∥f∥p<∞ L p T 1 T 2 f T 1 T 2 p f ∥f∥p=∫ T 1 T 2 |ft|pdt1p p f t T 2 T 1 f t p 1 p where 1≤p<∞ 1 p ∥f∥p=esssup|ft| p f ess sup f t where T 1 ≤t≤ T 2 T 1 t T 2
Problem 3
What does it take for and f f to be in L p T 1 T 2 L p T 1 T 2 ?

Continuous-Time and Infinite-Length

figure8.png
Figure 10
We will still refer to ft f t as a vector.
Definition 4:
L p R= f∥f∥p<∞ L p R f p f ∥f∥p=∫-∞∞|fn|pdt1p p f t f n p 1 p where 1≤p<∞ 1 p ∥f∥∞=esssup|ft| f ess sup f t where -∞<t<∞ t
Problem 4
What does it take for an f∈ L p R f L p R ?
Example 6 
Sketch an f∈ L p R f L p R and f∉ L p R f L p R .

Power

What do we do when ∥f∥p=∞ p f ?
Example 7: Periodic Signal 
figure9.png
Figure 11
Solution: Look at the "norm per unit time".
  • ie - norm over one period.
  • ie - norm of infinite-length signal converted to finite length signal by windowing.
    figure10.png
    Figure 12
    ∥f∥pT p f T is the measure.
Units for p=2 p 2 ?
L 2 L 2 Power = "energy per unit time"
  • Useful when E f =∞ E f
  • time average of energy
P f =limT→∞∫-T2T2|ft|2dt P f T T t T 2 T 2 f t 2
figure11.png
Figure 13
  1. compute EnergyT=∥f∥22T Energy T 2 f 2 T
  2. look at limT→∞EnergyT=∥f∥22T T Energy T 2 f 2 T
P f P f is often called the mean-square value of f f. P f P f is called the root mean squared (RMS) value of f f.
Units?
"Energy signals" have finite norm (energy) E f <∞ E f .
"Power signals" have finite and nonzero power P f <∞ P f , P f ≠0 P f 0 , and ( → E f =∞ E f ).

Conclusions

Energy signals are not power signals.
Power signals are not energy signals.
Why?
Problem 5
Are all signals either energy or power signals?
Example 8 
ft=t f t t
figure12.png
Figure 14
The 4 fundamental classes of signals we will study depend on the independent (time) variable.
figure13.png
Figure 15

Comments, questions, feedback, criticisms?

Send feedback