Skip to content Skip to navigation

Connexions

You are here: Home » Content » Signal Representation

Navigation

Content Actions
  • Download module PDF
  • Add to ...
    Add the module to:
    • My Favorites
    • A lens
    • An external social bookmarking service
    • My Favorites (What is 'My Favorites'?)
      'My Favorites' is a special kind of lens which you can use to bookmark modules and collections directly in Connexions. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need a Connexions account to use 'My Favorites'.
    • A lens (What is a lens?)

      Definition of 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.

    • External bookmarks
  • E-mail the authors

Recently Viewed

Signal Representation

Module by: Feng Qiao, Rachael Milam

The idea of signal representation is to decompose a general signal into a linear combination of a fixed set of vectors.

Example 1

For any v∈ℝ2 v 2 , we have v=x10+y01 v x 1 0 y 0 1

Example 2

Suppose we have an orthonormal set of vectors v k t v k t which spans a subspace VV of L2ℝ L 2 , then ∀,ft∈V f t V , we have

ft=∑k<ft, v k t> v k t f t k k f t v k t v k t (1)

In practical applications, signals are generally smooth or at least piecewise smooth (e.g., sound waves, image intensities). Such signals can be well approximated by using piecewise polynomials.

A function is piecewise polynomial with K+1 K 1 pieces if

pt=∑i=0K p i twt p t i 0 K p i t w t (2)
wt=1if t i ≤t< t i + 1 0otherwise w t 1 t i t t i + 1 0 where t 0 =0 t 0 0 and t K + 1 =T t K + 1 T and p i t=∑d=0D a i , d td p i t d 0 D a i , d t d are polynomials of max degree DD.

So when we try to project a polynomial signal ft=tn f t t n to a subspace spanned by v k t v k t , we will get

f ^ t=∑k<tn, v k t> v k t f ^ t k k t n v k t v k t (3)
The term <tn, v k t>=∫tn v k tdt t n v k t t t n v k t is called the nth n th moment of v k t v k t . In many cases we would like a "parsimonious" representation of the signal (e.g., compression of the signal). That is, we would like most of the coefficients to be zero. In the problem of polynomial signal representation, that is to say, we want the moments of the basis vector to be zero. This is called vanishing moments of v k t v k t .

question:

How to choose v k t v k t wisely so that we have as many vanishing moments as possible?

Comments, questions, feedback, criticisms?

Send feedback