Skip to content Skip to navigation Skip to collection information

OpenStax_CNX

You are here: Home » Content » Digital Signal Processing (Ohio State EE700) » Haar Approximation at the kth Coarseness Level

Navigation

Table of Contents

Recently Viewed

This feature requires Javascript to be enabled.
 

Haar Approximation at the kth Coarseness Level

Module by: Phil Schniter. E-mail the author

It is instructive to consider the approximation of signal xt 2 x t 2 at coarseness-level-kk of the Haar system. For the Haar case, projection of xt 2 x t 2 onto V k V k is accomplished using the basis coefficients

c k , n = φ k , n txtdt=n2k(n+1)2k2k2xtdt c k , n t φ k , n t x t t n 2 k n 1 2 k 2 k 2 x t
(1)
giving the approximation
x k t=n= c k , n φ k , n t=n=n2k(n+1)2k2k2xtdt φ k , n t=n=12kn2k(n+1)2kxtdt(2k2 φ k , n t) x k t n c k , n φ k , n t n t n 2 k n 1 2 k 2 k 2 x t φ k , n t n 1 2 k t n 2 k n 1 2 k x t 2 k 2 φ k , n t
(2)
where 12kn2k(n+1)2kxtdt=average value of x(t) in interval 1 2 k t n 2 k n 1 2 k x t average value of x(t) in interval 2k2 φ k , n t=height=1   k 2 k 2 φ k , n t height 1 This corresponds to taking the average value of the signal in each interval of width 2k 2 k and approximating the function by a constant over that interval (see Figure 1).

Figure 1
Figure 1 (haarapprox_Vk.png)

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

Collection 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

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