<< Chapter < Page Chapter >> Page >

Given a mother scaling function φ t 2 —the choice of which will be discussed later—let us construct scaling functions at"coarseness-level- k" and "shift- n " as follows: φ k , n t 2 k 2 φ 2 k t n . Let us then use V k to denote the subspace defined by linear combinations of scaling functions at the k th level: V k span φ k , n t n . In the Haar system, for example, V 0 and V 1 consist of signals with the characteristics of x 0 t and x 1 t illustrated in .

We will be careful to choose a scaling function φ t which ensures that the following nesting property is satisfied: V 2 V 1 V 0 V -1 V -2 coarse detailed In other words, any signal in V k can be constructed as a linear combination of more detailed signals in V k 1 . (The Haar system gives proof that at least one such φ t exists.)

The nesting property can be depicted using the set-theoretic diagram, , where V 1 is represented by the contents of the largest egg (which includes the smaller two eggs), V 0 is represented by the contents of the medium-sized egg (which includes the smallest egg), and V 1 is represented by the contents of the smallest egg.

Going further, we will assume that φ t is designed to yield the following three important properties:

  • φ k , n t n constitutes an orthonormal basis for V k ,
  • V 0 (contains no signals).
    While at first glance it might seem that V should contain non-zero constant signals ( e.g. , x t a for a ), the only constant signal in 2 , the space of square-integrable signals, is the zero signal.
  • V 2 (contains all signals).
Because φ k , n t n is an orthonormal basis, the best (in 2 norm) approximation of x t 2 at coarseness-level- k is given by the orthogonal projection,
x k t n c k , n φ k , n t
c k , n φ k , n t x t

We will soon derive conditions on the scaling function φ t which ensure that the properties above are satisfied.

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Digital signal processing (ohio state ee700). OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10144/1.8
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Digital signal processing (ohio state ee700)' conversation and receive update notifications?

Ask