variable [ 13 ,
13Definition 3.1] and by a relation between M(f ) andM (f ) shown in [17, Eq. (5)]. By using the phase φ(f ) ofM (f ), we can rewrite the complementary PSD asM (f ) = |M(f )|e jφ(f ) = k(f ) M(f )M(−f )e jφ(f ) , where 0 ≤ φ(f ) ≤ 2π. In the next lemma, the properties of the impropriety frequency and the phase functions are provided. Lemma 2: The impropriety frequency function k(f ) and the phase function φ(f ) satisfy 0 ≤ k(f ) ≤ 1, k(−f ) = k(f ), and φ(−f ) = φ(f ), ∀f. (17) Proof: Sincem[−k] =m[k] by definition, we haveM (−f ) =M (f ), which implies φ(−f ) = φ(f ). This also leads to k(−f ) = k(f ) by (16). By using the property |M(f )| 2 ≤ M(f )M(−f ) shown in [17, Eq. (5)], we have 0 ≤ k(f ) ≤ 1. ✷ For example, an uncorrelated real-valued PAM **data** sequence results in k(f ) = 1, ∀f , whereas any proper-complex **data** sequence results in k(f ) = 0, ∀f . By using the impropriety frequency function, we can rewrite the MSE (15) in the form of a function ofs(f ) as a function of s(f ).