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Table 1 Various topics and applications with sparsity properties: the sparsity, which may be in the time/space or “frequency” domains, consists of unknown samples/coefficients that need to be determined

From: A unified approach to sparse signal processing

 

0

1

2

3

4

5

6

7

8

1

Category

Topics

Sparsity

Type of

Information

Type of

Min number

Conventional

Applications

   

domain

sparsity

domain

sampling in

of required

reconstruction

 
      

info. domain

samples

methods

 

2

Sampling

Uniform

Frequency

Lowpass

Time/space

Uniform

BW−1

Lowpass

A/D

  

sampling

     

filtering/

 
        

Interpolation

 

3

 

Nonuniform

Frequency

Lowpass

Time/space

Missing samp-

BW−1

Iterative metho-

Seismic/

  

sampling

   

-les/jitter/per-

(in some cases

-ds/filter banks/

MRI/CT/

      

-iodic/random

even BW)

spline interp.

FM/ PPM

4

 

Sampling of

Frequency

Union of

Time/pace

Uniform/jit-

2×BW

Iterative metho-

Data

  

multiband

 

disjoint

 

-ter/periodic/

 

-ds/filter banks/

compression/

  

signals

 

intervals

 

random

 

interpolation

radar

5

 

Random

Frequency

Random

Time/space

Random/

2×#

Iterative methods:

Missing samp.

  

sampling

   

uniform

coeff.

adapt. thresh.

recovery/

        

RDE/ELP

data comp.

6

 

Compressed

An arbitrary

Random

Random

Random

c·k·log( n k )

Basis pursuit/

Data

  

sensing

orthonormal

 

mapping of

mixtures

 

matching

compression

   

transform

 

time/space

of samples

 

pursuit

 

7

 

Finite

Time and

Random

Filtered

Uniform

# Coeff. + 1 +

Annihilating

ECG/

  

rate of

polynomial

 

time

 

2·(# discont.

filter

OCT/

  

innovation

coeff.

 

domain

 

epochs)

(ELP)

UWB

8

Channel

Galois

Time

Random

Syndrome

Uniform

# errors

Berlekamp

Digital

 

coding

field

   

or

 

-Massey/Viterbi/

communic-

  

codes

   

random

 

belief prop.

-tion

9

 

Real

Time

Random

Transform

Uniform

# impulsive

Adaptive

Fault

  

field

  

domain

or

noise

thresholding

tolerant

  

codes

   

random

 

RDE/ELP

system

10

Spectral

Spectral

Frequency

Random

Time/

Uniform

# tones

MUSIC/

Military/

 

estimation

estimation

  

autocor-

 

−1

pisarenko/

radars

     

-relation

  

prony/MDL

 

11

Array

MSL/

Space

Random

Space/

Uniform

MDL+

Radars/

 

processing

DOA

  

autocor-

 

# sources

MUSIC/

sonar/

  

estimation

  

-relation

  

ESPRIT

ultrasound

12

 

Sparse arr-

Space

Random/

Space

Peaks of

# desired

Optimiz-

Radars/sonar/

  

-ay beam-

 

missing

 

sidelobes/

array

-ation: LP/

ultrasound/

  

-forming

 

elements

 

[non]uniform

elements

SA/GA

MSL

13

 

Sensor

Space

Random

Space

Uniform

2× BW

Similar

Seismic/

  

networks

    

of random

to row 5

meteorology/

       

field

 

environmental

14

SCA

BSS

Active

Random

Time

Uniform

# active

l /2/

Biomedical

   

source/time

   

sources

SL0

 

15

 

SDR

Dictionary

Uniform/

Linear mix-

Random

# sparse

l /2/

Data compression

    

random

-ture of time

 

words

SL0

 
     

samples

    

16

Channel

Multipath

Time

Random

Frequency

Uniform/

# Spa-

l /

Channel equaliz-

 

estimation

channels

  

or time

nonuniform

-rse channel

MIMAT

-ation/OFDM

       

components

  
  1. The information domain consists of known samples/coefficients in the “frequency” or time/space domain (the complement of the sparse domain). A list of acronyms is given in Table 2 at the end of this section; also, a list of common notations is presented in Table 3. For definition of ESPRIT on row 11 and column 7, see the footnote on page 41.