A motivating application sensor array signal processing
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Forward. Inverse. A Motivating Application: Sensor Array Signal Processing. Goal: Estimate directions of arrival of acoustic sources using a microphone array. Data collection setup. Underlying “sparse” spatial spectrum f *. [. ]. Underdetermined Linear Inverse Problems.

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A Motivating Application: Sensor Array Signal Processing

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A motivating application sensor array signal processing

Forward

Inverse

A Motivating Application:Sensor Array Signal Processing

  • Goal: Estimate directions of arrival of acoustic sources using a microphone array

Data collection setup

Underlying “sparse” spatial spectrum f*


Underdetermined linear inverse problems

[

]

Underdetermined Linear Inverse Problems

  • Basic problem: find an estimate of , where

  • Underdetermined -- non-uniqueness of solutions

  • Additional information/constraints needed for a unique solution

  • A typical approach is the min-norm solution:

  • What if we know is sparse (i.e. has few non-zero elements)?


Sparsity constraints

Number of non-zero elements in f

Sparsity constraints

  • Prefer the sparsest solution:

  • Can be viewed as finding a sparse representation of

    the signal y in an overcomplete dictionary A

  • Intractable combinatorial optimization problem

  • Are there tractable alternatives that might produce the same result?

  • Empirical observation:l1-norm-based techniques produce solutions that look sparse

    • l1 cost function can be optimized by linear programming!


L 1 norm and sparsity a simple example

l1-norm and sparsity – a simple example

A sparse signal

1.4142

2.0000

A non-sparse signal

0.5816

3.5549

  • Goal: Rigorous characterization of the l1 – sparsity link

For these two signals f1 and f2 we have A*f1=A*f2 where A is a 16x128 DFT operator


L 0 uniqueness conditions

Number of non-zero elements in f

  • Thm. 1:

  • What can we say about more tractable formulations like l1 ?

where

and K(A) is the largest integer such that any set of K(A) columns of A is linearly independent.

Unique l0solution

l0 uniqueness conditions

  • Prefer the sparsest solution:

  • Let where

  • When is ?


L 1 equivalence conditions

  • Thm. 2(*):

    • is sparse enough exact solution by l1 optimization

  • Can solve a combinatorial optimization problem by convex optimization!

where

(*) Donoho and Elad obtained a similar result concurrently.

l1solution = l0solution !

l1 equivalence conditions

  • Consider the l1 problem:

  • Can we ever hope to get ?


L p p 1 equivalence conditions

  • Thm. 3:

where

lpsolution = l0solution !

  • Smaller p  more non-zero elements tolerated

  • As p0 we recover the l0 condition, namely

Smaller p

lp (p ≤ 1) equivalence conditions

  • Consider the lp problem:

  • How about ?


L 0 uniqueness conditions1

Number of non-zero elements in f

  • Definition: The index of ambiguity K(A) of A is the largest integer such that any set of K(A) columns of A is linearly independent.

  • Thm. 1:

  • What can we say about more tractable formulations like l1 ?

Unique l0solution

l0 uniqueness conditions

  • Prefer the sparsest solution:

  • Let

  • When is ?


L 1 equivalence conditions1

  • Definition: Maximum absolute dot product of columns

  • Thm. 2(*):

    • is sparse enough exact solution by l1 optimization

  • Can solve a combinatorial optimization problem by convex optimization!

(*) Donoho and Elad obtained a similar result concurrently.

l1solution = l0solution !

l1 equivalence conditions

  • Consider the l1 problem:

  • Can we ever hope to get ?


L p p 1 equivalence conditions1

  • Definition:

  • Thm. 3:

Smaller p

lpsolution = l0solution !

  • Smaller p  more non-zero elements tolerated

  • As p0 we recover the l0 condition, namely

lp (p ≤ 1) equivalence conditions

  • Consider the lp problem:

  • How about ?


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