Skip to content Skip to navigation

OpenStax_CNX

You are here: Home » Content » Why Eigenstuff Matters in DSP

Navigation

Recently Viewed

This feature requires Javascript to be enabled.
 

Why Eigenstuff Matters in DSP

Module by: Richard Baraniuk. E-mail the author

Summary: An explaination as to why eigenstuff matters in DSP

Consider processing a signal xCN x N with system represented by a normal matrix A A

Figure 1
Figure 1 (fig1.png)

y=Ax y A x
(1)
  • matrix multiply
  • yn=K=0N1xk A n , k y n K 0 N 1 x k A n , k
    (2)
    yn y n involves coupling all xk x k via A n , k A n , k
  • cost: ON2 O N 2 in general
  • one-step procedure

Now consider representation

A=VΛVH A V Λ V
(3)
Note that the superscript H indicates the matrix conjugate transpose (also called the Hermitian transpose). This is implemented by a three-step procedure:
y=Ax=VΛVHx y A x V Λ V x
(4)
  1. α=VHx α V x This computes the coefficients α α that build up x x in terms of eigenbasis v i v i : x=Vα x V α (ie: VH V takes x x from the "time-domain" to the "eigenbasis domain".
  2. Λα Λ α with Λ Λ a diagonal matrix Λ=( λ 0 000 0 λ 1 00 00 ... ... 0 00... λ N - 1 ) Λ λ 0 0 0 0 0 λ 1 0 0 0 0 ... ... 0 0 0 ... λ N - 1 ie: we just do element-wise multiplication λ i α i λ i α i (super easy and no coupling between α i α i 's)
  3. V(Λα)=y V Λ α y takes the vector Λα Λ α back to the time-domain.

Hallmarks

  • Once we transform x x to the eigenbasis domain of A A, we can perform an operation equivalent to A A but independently on each α i α i .
  • This works for any normal matrix A A.
  • Most of the systems studied in DSP are "linear shift-invariant":
    • all correspond to normal matrices
    • all share the same V V: DFT basis
All LSI processing y=Ax y A x can be done through:
  1. α=fftx α fft x
  2. α ~ =λα α ~ λ α where λ λis diagonal of eigen matrix Λ Λ
  3. y=ifft α ~ y ifft α ~
The Cost:
  • Step 1: ONlogN O N N
  • Step 2: ON O N
  • Step 3: ONlogN O N N
  • total: ONlogN O N N
ONlogN<ON2 O N N O N 2
(5)
y=Ax y A x directly!!!

Useful Formulas

A=VΛVH=( v 0 v 1 v 2 ... v N - 1 )( λ 0 000 0 λ 1 00 00 ... ... 0 00... λ N - 1 )( VH 0 VH 1 VH 2 ... VH N - 1 ) A V Λ V v 0 v 1 v 2 ... v N - 1 λ 0 0 0 0 0 λ 1 0 0 0 0 ... ... 0 0 0 ... λ N - 1 V 0 V 1 V 2 ... V N - 1
(6)

Content actions

Download 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 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