Table 1 FAWC optimization

Objective: Measurement matrix optimization with

given dictionary D, i.e.,

\({\boldsymbol {\Phi }} =\mathop {\arg \min }\limits _{\boldsymbol {\Phi }} \left \{ {\frac {1}{2}\eta + \left ({1 - \alpha } \right)\mu _{NC}^{t} + \alpha {\mu _{C}^{t}}} \right \}.\)

Initialization:Set n=0, and calculate the

eigenvalue decomposition of D D ^H, i.e.,

D D ^H=U Λ U ^H.

Then, we calculate the initial value of Φ

according to

\({{\boldsymbol {\Phi }}^{\left (0 \right)}} = \left [ {\begin {array}{cccc} {{{\mathbf {I}}_{M}}}&0 \end {array}} \right ]{{\boldsymbol {\Lambda }}^{- \frac {1}{2}}}{{\mathbf {U}}^{H}}.\)

Repeat:

a). Update G ⁽ⁿ⁾ according to Φ ⁽ⁿ⁾:

G ⁽ⁿ⁾=(Φ ⁽ⁿ⁾ D)^H Φ ⁽ⁿ⁾ D.

b). Calculate h _t (G ⁽ⁿ⁾) according to the

forthcoming Eq. (24), and form the matrix

\( {\boldsymbol {\Upsilon }=} {{\boldsymbol {\Lambda }}^{- \frac {1}{2}}}{{\mathbf {U}}^{H}}{\mathbf {D}}{h_{t}}\left ({{{\mathbf {G}}^{\left (n \right)}}} \right){\mathbf {U}}{{\mathbf {D}}^{H}}{{\boldsymbol {\Lambda }}^{- \frac {1}{2}}}.\)

c). Calculate the eigenvalue decomposition of Υ,

and find its M top eigenvalues Δ _M and the

corresponding eigenvectors V _M of Υ.

d). Update measurement matrix according to

\({{\boldsymbol {\Phi }}^{\left ({n+1} \right)}} = {\boldsymbol {\Delta }}_{M}^{\frac {1}{2}}{\mathbf {V}}_{M}^{H}{{\boldsymbol {\Lambda }}^{- \frac {1}{2}}}{{\mathbf {U}}^{H}}.\)

e). n=n+1.

Until: Convergence criterion is satisfied.