Nonconvex Optimization via Joint Norm Relaxed SQP and Filled Function Method with Application to Minimax Two-Channel Linear Phase FIR QMF Bank Design - Advances in Heuristic Signal Processing and Applications - page 5

Let us denote

2 0 , 0 , 0 , cos N ω , cos N ω ,..., cos 3 ω T

− 1 − 5 − N

≡

κ(ω) , (1.16)

2 2 2

then

= η( 2 ω) T ¯ + e − jω(N − 1 ) η( 2 ω) + ¯ ,

H 0 (ω) (1.17)

therefore

× 0 , 0 , 0 ,e jω N − 2 ,e jω N − 2 ,...,e − jω N − 2 ¯
+ 0 , 0 , 0 ,e − jω N − 2 ,e − jω N − 2 ,...,e jω N − 2 ¯ .

e − jω N − 1

H 0 (ω) =

2

(1.18)

Finally,

H 0 (ω) = e − jω N − 2 κ(ω) T ¯ ,

(1.19)

and the passband ripple magnitude of the prototype filter can be expressed as
|

(κ(ω)) T ¯ −

|∀ ∈

D(ω) ω B p . Let us define

T ,

ι p ≡[ 0 , 1 , 0 ,..., 0 ] (1.20)

then the constraint on the maximum passband ripple magnitude of the prototype
filter can be expressed as
κ(ω) T ¯ −

D(ω) ≤ ι p ¯ ∀ ω ∈ B p . (1.21)

Let us define

¯ p (ω) ≡ κ(ω) − ι p , − κ(ω) − ι p T

(1.22)

and

≡ D(ω), D(ω) T , ¯ p (ω) − (1.23)

then the constraint on the maximum passband ripple magnitude of the prototype
filter can be further expressed as

¯ p (ω) ¯ − ¯ p (ω) ≤ ¯ ∀ ω ∈ B p . (1.24)

Similarly, let us define

T ,

ι s ≡[ 0 , 0 , 1 , 0 ,..., 0 ] (1.25)

≡ κ(ω) ι s T ,

¯ s (ω) − ι s , − κ(ω) − (1.26)

and

≡ D(ω), D(ω) T ,

¯ s (ω) − (1.27)