Finite Impulse Response Filters - Introduction to Digital Signal Processing and Filter Design - page 282

where b [ k ] = 2 h { [ ( 2 M + 1 )/ 2] − k } , 1 ≤ k ≤ [ ( 2 M + 1 )/ 2]. This can be fur-

ther reduced to the form

cos ω
2

( 2 M − 1 )/ 2

b [ k ]cos (kω)

H R (ω) = (5.54)

k = 0

where

2 b [1]

2 b [0]

1

b [1] = +

2 b [ k ]

1 ) ,

1 2 M − 1

2

+ b(k b [ k ] = − 2 ≤ k ≤ (5.55)

b 2 M

2 b 2 M

+ 1
2

1 − 1

2

=

Let us consider the function H R (ω) for a type III filter. Equation (5.49) can be
reduced to the form

( 2 M + 1 )/ 2

H R (ω) = c [ k ]sin (kω) (5.56)

k = 1

where c [ k ] = 2 h [ M − k ], 1 ≤ k ≤ M . This can be reduced to the form

M − 1

H R (ω) = sin (ω)

0

c [ k ]cos (kω) (5.57)

k =

where

= c [1]

1
2

c [1] c [0] −

1

c [ k ] =

2 (

c [ k − − c [ k ] ) , ≤ k ≤ M −

1] 2 1 (5.58)

1
2

c [ M ] = c [ M − 1]

Finally we express H R (ω) for a type IV filter as

( 2 M + 1 )/ 2

d [ k ]sin k

2 ω

1

H R (ω) = − (5.59)

k = 1

where d [ k ] = 2 h { [ ( 2 M + 1 )/ 2] − k } ,1 ≤ k ≤ ( 2 M + 1 )/ 2. Equation (5.59) can

be reduced to the form

= sin ω

2

( 2 M − 1 )/ 2

d [ k ]cos (kω)

H R (ω) (5.60)

k = 0