v2009.01.01 - Convex Optimization

4.6. CARDINALITY AND RANK CONSTRAINT EXAMPLES 331
We can cut that number of constraints in half via vertical and horizontal
mask Φ symmetry which forces the imaginary inverse transform to 0 : The
inverse subsampled transform in matrix form is
and in vector form
later abbreviated
where
F H (ΘΦΘ ◦ F UF)F H = F H KF H (756)
(F H ⊗F H )δ(vec ΘΦΘ)(F ⊗F ) vec U = (F H ⊗F H ) vec K (757)
P vec U = f (758)
P ∆ = (F H ⊗F H )δ(vec ΘΦΘ)(F ⊗F ) ∈ C n2 ×n 2 (759)
Because of idempotence P = P 2 , P is a projection matrix. Because of its
Hermitian symmetry [140, p.24]
P = (F H ⊗F H )δ(vec ΘΦΘ)(F ⊗F ) = (F ⊗F ) H δ(vec ΘΦΘ)(F H ⊗F H ) H = P H
(760)
P is an orthogonal projector. 4.47 P vec U is real when P is real; id est, when
for positive even integer n
[
]
Φ
Φ = 11 Φ(1, 2:n)Ξ
∈ R n×n (761)
ΞΦ(2:n, 1) ΞΦ(2:n, 2:n)Ξ
where Ξ∈ S n−1 is the order-reversing permutation matrix (1608). In words,
this necessary and sufficient condition on Φ (for a real inverse subsampled
transform [246, p.53]) demands vertical symmetry about row n +1 and
2
horizontal symmetry 4.48 about column n+1.
2
Define ⎡
⎤
1 0 0
−1 1 0
∆ =
∆ −1 1 . ..
∈ R ...
...
...
n×n (762)
⎢
⎣
...
⎥
1 0 ⎦
0 T −1 1
4.47 (759) is a diagonalization of matrix P whose binary eigenvalues are δ(vec ΘΦΘ) while
the corresponding eigenvectors constitute the columns of unitary matrix F H ⊗F H .
4.48 This condition on Φ applies to both DC- and Nyquist-centric DFT matrices.