5.7.3 Causal Generalized Linear-Phase FIR Systems.

· h(n)=0, n<0 and n>M

(1) Symmetric impulse response· b=0 or p이면

[ ]

[ ][ ] [ ] 0)M(sin)M(h)1M(sin)1M(h

)1(sin)1(h)sin()0(h

0)n(sin)n(hM

0n

=a-w+a--w-++a-w+aw-®

=a-w×å=

L

L

· 대칭조건에서 2a=M(integer)이면 sin[w(M-a)]=sinaw이고

sin[w(M-1-a)]=sin[w(a-1)]이므로, 위의식이만족하려면,

h(0)=h(M), h(1)=h(M-1), …… 이어야하므로

ççè

æ ££-=®

otherwise,0Mn0),nM(h

)n(h

ççè

æ==

×egerintoddMegerintevenM

responseimpuseSymmetric

2Mj

e)e(A)e(H je

j w-ww =× 1

(2) Antisymmetric impulse response

[ ]

[ ][ ] [ ] 0)M(cos)M(h)1M(cos)1M(h

)1(cos)1(h)cos()0(h

0)dn(cos)n(h

23or

2M

0n

=a-w+a--w-++a-w+aw®

=-w

pp=b×

å=

L

L

· 대칭조건에서 2a=M을고려하면

h(0) = -h(M), h(1) = -h(M-1), …… 이어야하므로

ççè

æ ££--=®

otherwise,0Mn0),nM(h

)n(h

2j

2Mj

2Mj

e)e(Ae)e(jA)e(H j0

j0

j pww +-w-ww ==×

• Antisymmetric impulse responseççè

æegerintodd:Megerinteven:M

2

(3) Impulse response of FIR linear-phase systems

impulse response of FIR linear-phase systems

(a) type I, M even, h(n) = h(M-n), (b) type II, M odd, h(n) = h(M-n)

(c) type III, M even, h(n)= -h(M-n), (d) type IV, M odd, h(n) = -h(M-n) 3

(4) Type I FIR linear-phase systems

å å

å-

= =

w--w-

=

w-w

+

w

++=

=×

££-=×

22M

0n

M

n

nj2Mnj

M

0n

njj

22M

2Mj

e)n(he)(he)n(h

e)n(h)e(H

egerintevenanMwithMn0),nM(h)n(h

위의 세번째항을변수치환하여 정리하면

( ) ( )( )

( )( ) ( ) ( )

( ) ( )

( ) ( )

0,2M

,,2,1m,mh2)m(andh)0(where

mcos)m(e

hmcosmh2e

nm,hncos)n(h2e

)(hee)n(he)e(H

e)nM(h

2M

2M

2M

0m

1m2M

2M

2M

0n2M

2M

2Mnjnj

0n

j

0n

)nM(j

2M

2Mj

2M

2Mj

22M

2Mj

2M

2M2

2M

2Mj

22M

=b=a×

=-==

úúû

ù

êêë

éw=

úúû

ù

êêë

é+w-=

-=úúû

ù

êêë

é+-w=

úúû

ù

êêë

é++=

-

å

å

å

å

å

=

-

=

-

=

-

-w--w

=

-w

=

-w-

w

w

-w

-w

-

Laa

a

되므로이

4

Frequency response of type I systems5

(5) Type II FIR linear-phase systems

( ) ( )( )

( )( ) ( )

( )( )

( )

불가능설계이므로에서 HPF 0)e(H

,,2,1m,mh2)m(bwhere

mcos)m(be

mn,ncos)n(h2e

ee)n(he

e)nM(he)n(h

)nMm(,e)n(he)n(h)e(H

egerintoddanMwithMn0),nM(h)n(h

j

21M

21M

1m21

21M

0n21

21M

0n

njnj

0n 0n

)nM(jnj

21M

0n

M

n

njnjj

21M

2Mj

21M

2Mj

21M

2M

2M

2Mj

21M

21M

21M

=p=w×

=-=

úúû

ù

êêë

é-w=

=-úúû

ù

êêë

é--w=

úúû

ù

êêë

é+=

-+=

-=+=×

££-=×

w

++

=

-

+

=

+-

=

-w-w-

= =

-w-w-

-

= =

w-w-w

å

å

å

å å

å å

+w

-w

-w

- -

+

L

6

frequency response of type II system 7

(6) Type III FIR linear-phase systems

( )

( )

( )

불가능설계이므로에서 HPF,LPF 0)e(H,0

,,2,1m,mh2)m(cwhere

msin)m(ceeeH

0h

egerintevenanMwithMn0),nM(h)n(h

j

2M

2M

1m

j

2M

2M

2j

2Mj

=p=w=w×

=-=

úúû

ù

êêë

éw=×

=×

££--=×

w

=

-w åpw

L

8

frequency response of type III system 9

(7) Type IV FIR linear-phase systems

( ) ( )( )

( )

불가능설계이므로에서 LPF 0)e(H,0

,,2,1m,mh2)m(dwhere

msin)m(deeH

egerintoddanMwithMn0),nM(h)n(h

j

21M

21M

21M

1m21j 2

Mj

==w×

=-=

úúú

û

ù

êêê

ë

é-w=×

££--=×

w

++

+

=

-w åw

L

10

frequency response of type IV system 11

(8) Location of zeros for FIR linear-phase systems

( )

( ) ( )[ ]

( ) ( )[ ]

. zero )z(H

er1z reciprocalrez zero H(z)

. zero )z(Hzero )z(H)z(HZ)z(H

ZZ)1(hZZ)0(hZ)z(H

ZZZZ)1(hZZ)0(hZ)z(H

ryantisymmet:,symmetric:whereZ)0(hZ)1(hZ)2(h

Z)2(hZ)1(h)0(h

Z)n(hzH

j1j

1

M1

1

1

M)1M()2M(

21

M

0n

n

2)2M(

22M

2M

2M

2M

2)2M(

22M

2M

2M

2M

된다가의

도관계인이면가의

같다는의와의

대입하면을대신

q--q

-

-

-

-

---

-----

--

=

-

==®

®

±=®

+±+±=

+±+±=×

-+±±±

+++=

=×

--

--

å

LL

LL

LL

LL

12

· h(n)이 real 일때의 H(z) zero 분류

i) r ¹ 1, q ¹ 0, p (단위원상에있지않는복소공액영점)

( )( )( )( )4321

1jr11j

r11j1j

zbzczbz

ze1ze1zre1zre1----

-q--q-q--q

++++=

----

aa

ii) r = 1, q ¹ 0, p (단위원상의복소공액영점)

( )( )1j1j ze1ze1 -q--q --

iii) r ¹ 1, q = 0, p (단위원상에있지않는실영점)

( )( )11 rz1rz1 -- ±±

iv) r = 1, q = 0, p (단위원상의실영점)

( )1z1 -±

13

· 위의 iv)의경우를고려한 type II, type III, type IV 의성질

i) Symmetric impulse response (type I, type II)

. 0)H(eH(-1) )1(H)1(H)IItype( M

.)Itype( M

)1(H)1()1(H )z(Hz)z(H

j

MM1

한다이어야되므로이경우기수인이

성립한다경우우수인이

이므로

==--=-®

®

--=-=

p

-

ii) Antisymmetric impulse response (type III, type IV)

( )

( ) ( )

. 0)H(e1)H( )1(H)1(H)IVtype( M

. 0eH)1(H0eH)1(H )1(H)1(H),IIItype( M

)1(H)1()1(H )z(HzzH

j0

j0j

MM1

한다되어야가

되므로이경우기수인이

한다만족되어야가와

되므로이경우우수인이

이므로

==+

+-=±®

==-==+

±-=±®

±±-=±-=

p

-

14

typical plots of zeros for linear-phase systems15

5.7.4 Relation of FIR Linear-Phase Systems to Minimum-Phase Systems

)z(HofzerosofnumbertheisMwhere)z(HZ)z(H

reciprocal zero )z(H zero )z(H)z(H)z(H)z(H)z(H

mini

1min

Mmax

maxmin

maxucmin

i --=

×=×

있으므로관계가는의와의

16