Finite Impulse Response Filter Design Equations by the Frequency Sampling Design Method - Digital Signal Processing: Fundamentals and Applications - page 819

To devise the design procedure, we substitute Equation (E.2) in Equation (E.8) to obtain

N e jðN 1 Þ U = 2 N 1

1

k ¼ 0 H k e j 4 k þjðN 1 Þk p =N sin ½Nð U 2 p k=NÞ= 2

Hðe j U Þ¼ (E.11)

sin ½ð U 2 p k=NÞ= 2 Þ

It is required that the frequency response of the designed FIR filter expressed in Equation (E.11) be
linear phase. This can easily be accomplished by setting

4 k þðN 1 Þk p =N ¼ 0 ; 0 k N 1 (E.12)

in Equation (E.11) so that the summation part becomes a real value, thus resulting in the linear phase of
Hðe j U Þ , since only one complex term, e jðN 1 Þ U = 2 , is left, which presents the constant time delay of the
transfer function. Second, the sequence hðnÞ must be real. To proceed, let N ¼ 2 M þ 1, and due to the
properties of DFT coefficients, we have

HðkÞ¼HðN kÞ; 1 k M (E.13)

where the bar indicates complex conjugate. Note the fact that

W k

N ¼ W ðNkÞ

; 1 k M (E.14)

N

From Equation (E.1) , we write

!

M
k ¼ 1 HðkÞW k N þ

X

2 M
k ¼Mþ 1 HðkÞW kn

1
N

hðnÞ¼ Hð 0 Þþ (E.15)

N

Equation (E.15) is equivalent to

!

M
k ¼ 1 HðkÞW k N þ

M
k ¼ 1 HðN kÞW ðNkÞn

1
N

hðnÞ¼ Hð 0 Þþ

N

Using Equations (E.13) and (E.14) in the last summation term leads to

!

M
k ¼ 1 HðkÞW k N þ

M
k ¼ 1 HðkÞW kn

1
N

hðnÞ¼ Hð 0 Þþ

N

!

M
k ¼ 1 ðHðkÞW k N þ HðkÞW k N Þ

1
2 M þ 1

¼ Hð 0 Þþ

Combining the last two summation terms, we achieve

( M

k ¼ 1 HðkÞW kn

!)

1
2 M þ 1

hðnÞ¼ Hð 0 Þþ 2Re ; 0 n N 1 (E.16)

N

Solving Equation (E.12) gives

4 k ¼ðN 1 Þk p =N;

0 k N 1 (E.17)