Cyber-Physical Systems

Signal and Data Processing Techniques for Industrial Cyber-Physical Systems 191

enough random measurements are collected. Formally, a signal s ∈ R

N

is called k-sparse if s

0

< k,

where s

0

= # nonzero elements of s. This signal can be reliably recovered from a low-dimensional

representation y = s ∈ R

M

, where M  N by solving an 

0

-constrained minimization problem

given by

min s

0

subject to y = s. (8.1)

To guarantee the stable recovery of the original signal, the M ×N sensing matrix  must satisfy

the so-called restricted isometry property (RIP). A sensing matrix  ∈ R

M×N

satisfies the RIP with

isometry constant 0 ≤ δ < 1 if for all k-sparse signals, s, it holds that

(1 −δ)s

2

2

≤x

2

2

≤ (1+δ)s

2

2

. (8.2)

Designing such a sensing matrix is proven to ...