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# 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

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)

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