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Size of A Signal: Norms

Module by: Richard Baraniuk. E-mail the author

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

Note: Your browser may not currently support MathML. See our browser support page for additional details. You can always view the correct math in the PDF version.

"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

Figure 1
Figure 1 (figure1.png)
E f =∫-∞∞|ft|2dt E f t f t 2 (1)

Example 1

Calculate E f E f for

Figure 2
Figure 2 (figure2.png)

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

Figure 3
Figure 3 (figure3a.png)

Exercise 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

Figure 4
Figure 4 (figure3.png)

Discrete-Time Energy

Figure 5
Figure 5 (figure4.png)
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

Figure 6
Figure 6 (figure5.png)
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.

Figure 7
Figure 7 (figure6a.png)
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

Figure 8
Figure 8 (figure6.png)
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 .

Exercise 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

Figure 9
Figure 9 (figure7.png)
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

Exercise 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

Figure 10
Figure 10 (figure8.png)
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

Exercise 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

Figure 11
Figure 11 (figure9.png)
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.
    Figure 12
    Figure 12 (figure10.png)
    ∥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
Figure 13
Figure 13 (figure11.png)
  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?

Exercise 5

Are all signals either energy or power signals?

Example 8

ft=t f t t

Figure 14
Figure 14 (figure12.png)

The 4 fundamental classes of signals we will study depend on the independent (time) variable.

Figure 15
Figure 15 (figure13.png)

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