Complex-Valued Matrix Differentiation: Techniques and Key ... - Unik

Acepted 11.09.2006 for publication in IEEE Transactions on Signal Processing 10TABLE VDERIVATIVES OF FUNCTIONS OF THE TYPE F (Z, Z ∗ )F ( Z , Z ∗) Differential d vec(F ) D Z F ( Z, Z ∗) D Z ∗ F ( Z, Z ∗)Z I NQ d vec(Z ) I NQ 0 NQ×NQZ T K N,Q d vec(Z) K N,Q 0 NQ×NQZ ∗ I NQ d vec(Z ∗ ) 0 NQ×NQ I NQZ H K N,Q d vec(Z ∗ ) 0 NQ×NQ K N,Q())ZZ T I N 2 + K N,N (Z ⊗ I N ) d vec(Z)(I N 2 + K N,N (Z ⊗ I N ) 0 N 2 ×NQZ T Z(I Q 2 + K Q,Q)(I Q ⊗ Z T ) d vec(Z )(I Q 2 + K Q,Q)(I Q ⊗ Z T )0 Q 2 ×NQZZ H (Z∗ )⊗ I N d vec(Z )+K N,N (Z ⊗ I N ) d vec(Z ∗ ) Z ∗ ⊗ I N K N,N (Z ⊗ I N )Z −1(− (Z T ) −1 ⊗ Z −1) d vec(Z ) −(Z T ) −1 ⊗ Z −1 0 N 2 ×N 2Z p∑ p ((Z T ) p−i ⊗ Z i−1) p∑ (d vec(Z )(Z T ) p−i ⊗ Z i−1) 0 N 2 ×N 2i=1i=1Z ⊗ Z (A(Z )+B(Z ))d vec(Z ) A(Z )+B(Z ) 0 N 2 Q 2 ×NQZ ⊗ Z ∗ A(Z ∗ )d vec(Z )+B(Z)d vec(Z ∗ ) A(Z ∗ ) B(Z)Z ∗ ⊗ Z ∗ (A(Z ∗ )+B(Z ∗ ))d vec(Z ∗ ) 0 N 2 Q 2 ×NQA(Z ∗ )+B(Z ∗ )Z ⊙ Z 2 diag(vec(Z ))d vec(Z ) 2 diag(vec(Z )) 0 NQ×NQZ ⊙ Z ∗ diag(vec(Z ∗ ))d vec(Z )+diag(vec(Z))d vec(Z ∗ ) diag(vec(Z ∗ )) diag(vec(Z ))Z ∗ ⊙ Z ∗ 2diag(vec(Z ∗ ))d vec(Z ∗ ) 0 NQ×NQ 2diag(vec(Z ∗ ))d vec(F )=vec((dZ 0 ) ⊗ Z 1 )+vec(Z 0 ⊗ dZ 1 ). From Theorem 3.10 in [1], it follows thatvec ((dZ 0 ) ⊗ Z 1 )=(I Q0 ⊗ K Q1,N 0⊗ I N1 )[(d vec(Z 0 )) ⊗ vec(Z 1 )]=(I Q0 ⊗ K Q1,N 0⊗ I N1 )[I N0Q 0⊗ vec(Z 1 )] d vec(Z 0 ), (22)and in a similar way it follows that: vec (Z 0 ⊗ dZ 1 ) = (I Q0 ⊗ K Q1,N 0⊗ I N1 )[vec(Z 0 ) ⊗ I N1Q 1] d vec(Z 1 ).Inserting the last two results into d vec(F ) gives:d vec(F )=(I Q0 ⊗ K Q1,N 0⊗ I N1 )[I N0Q 0⊗ vec(Z 1 )] d vec(Z 0 )+(I Q0 ⊗ K Q1,N 0⊗ I N1 )[vec(Z 0 ) ⊗ I N1Q 1] d vec(Z 1 ). (23)Define the matrices A(Z 1 ) and B(Z 0 ) by A(Z 1 ) (I Q0 ⊗ K Q1,N 0⊗ I N1 )[I N0Q 0⊗ vec(Z 1 )], and B(Z 0 )=(I Q0 ⊗ K Q1,N 0⊗ I N1 )[vec(Z 0 ) ⊗ I N1Q 1]. It is then possible to rewrite the differential of F (Z 0 , Z 1 )=Z 0 ⊗ Z 1as d vec(F )=A(Z 1 )d vec(Z 0 )+B(Z 0 )d vec(Z 1 ). From d vec(F ), the differentials and derivatives of Z ⊗ Z,Z ⊗ Z ∗ , and Z ∗ ⊗ Z ∗ can be derived and these results are included in Table V. In the table, diag(·) returns thesquare diagonal matrix with the input column vector elements on the main diagonal [19] and zeros elsewhere.2) Moore-Penrose Inverse Related Problems: In pseudo-inverse matrix based receiver design, the Moore-Penrose inverse might appear [15]. This is applicable for MIMO, CDMA, and OFDM systems.Let F : C N×Q × C N×Q → C Q×N be given by F (Z, Z ∗ )=Z + , where Z ∈ C N×Q . The reason for includingboth variables Z and Z ∗ in this function definition is that the differential of Z + , see Proposition 1, depends onboth dZ and dZ ∗ . Using the vec(·) operator on the differential of the Moore-Penrose inverse in Table II, in additionto Lemma 4.3.1 in [18] and the definition of the commutation matrix, result in:[ (Z+d vec(F )=− ) ] [(T ⊗ Z+d vec(Z)+ I N − ( Z +) ) T ZT⊗ Z + ( Z +) ] HK N,Q d vec(Z ∗ )