On the Uniqueness of the Sparse Signals Reconstruction Based on the Missing Samples Variation Analysis

Algorithm 2

Uniqueness check—Theorem 3 and Corollary 4.

Require:

 (i) Set of missing sample positions

 (ii) Set of nonzero values in the reconstructed DFT

 (iii) Total number of signal samples ,

(1)

(2)

(3)

(4) for do

(5) for do

(6)    and

(7) end for

(8)  

(9) for do

(10)   and

(11) end for

(12) Sort array in non-decreasing order

(13) 

(14) if then                  Theorem 3 check

(15)  

(16) end if

(17) if then                  Corollary 4 check

(18)  

(19) end if

(20) end for

(21)

Output:

 (i) when the considered solution is unique.

 (ii) when the considered solution is unique with probability one excluding zero-probability event

  (when amplitudes of the signal components are related to each other with a relation defined by

  missing sample positions).

 (iii) when the considered solution is not unique.