The z-Transform of Symmetric Sequences

4.6 The z-Transform of Symmetric Sequences

In this section we explore some features of the phase characteristic and pole-zero pattern for systems having (anti-)symmetric impulse responses of finite duration.

4.6.1 Symmetric Sequences

Let us consider a symmetric sequence g[n] of duration N + 1 such that

g [n] = g[N − n] for n = 0 : N

< Case 1> If N is even, i.e., N = 2M for some integer M, then the z-transform of

g [n] is

4.6 The z-Transform of Symmetric Sequences 237

M−1 N

G [z] =

g [n]z −n

+ g[M]z −M +

g [n]z −n

n=M+1 (4.6.1)

n=0

= M−1

g [n](z −n

+z −(N −n) ) + g[M]z −M =z M−1 −M g [M] + g [n](z −n+M +z n−M

n=0

with M = 2 (4.6.2a)

n=0

which, with z = e jΩ , yields the frequency response as

G (Ω) = g [ ]+

N/ 2−1

2g[n] cos

2 −n ∠− Ω 2 (4.6.2b)

2 n=0

< Case 2> If N is odd, i.e., N = 2M − 1 for some integer M, then the z-transform of g[n] is

g [n]z [z] = −n

M−1

G g [n]z

g [n]z −n

= M−1 g [n](z −n

+z n−N/2 ) (4.6.3a) which, with z = e jΩ , yields the frequency response as

M=(N +1)/2

= M−1

z −N/2

g [n](z −n+N/2

2g[n] cos −n n=0 Ω ∠− Ω (4.6.3b)

2 2 Note that G[z]

| jΩ

−1 = G[e ] Ω =π = 0.

4.6.2 Anti-Symmetric Sequences

Let us consider an anti-symmetric sequence g[n] of duration N + 1 such that

(4.6.4) < Case 1> If N is even, i.e., N = 2M for some integer M, then we have g[M] =

g [n] = −g[N − n] for n = 0 : N

−g[M], which implies g[M] = 0. The z-transform of g[n] is

G [z] = z −M

M−1

g [n](z −n+M −z n−M ) with M = (4.6.5a)

n=0

238 4 The z-Transform which, with z = e jΩ , yields the frequency response as

G (Ω) =

N/ 2−1

2g[n] sin

−n Ω ∠− n=0 Ω 2 2 + (4.6.5b)

2 Note that G[e jΩ ]

Ω =0 = 0 and G[e ] Ω =π = 0. < Case 2> If N is odd, i.e., N = 2M − 1 for some integer M, then the z-transform

jΩ

of g[n] is

−z n−N/2 ) (4.6.6a) which, with z = e jΩ , yields the frequency response as

G [z] = z −N/2

M−1

g [n](z −n+N/2

n=0

(N −1)/2

2 + 2 Note that G[e jΩ ] Ω =0 = 0.

G (Ω) =

2g[n] sin

Remark 4.8 Pole-Zero Pattern and Linear Phase of (Anti-)Symmetric Sequences (1) From Eqs. (4.6.2a)/(4.6.5a) and (4.6.3a)/(4.6.6a), we can see that if G[z 0 ] = 0,

then G[z −1 0 ] = 0, which implies that real zeros occur in reciprocal pairs (z 0 and z −1 0 ) and complex zeros occur in reciprocal, complex-conjugate quadruplets (z 0 =r 0 ∠±Ω 0 and z −1 0 =r 0 −1 ∠±Ω 0 ). Note that zeros on the unit circle form their own reciprocal pairs and real zeros on the unit circle, i.e., z = 1 or z = −1, form their own reciprocal, complex conjugate pairs. Note also that all the poles are located at z = 0 or ∞. See Fig. 4.9(a1) and (a2).

(2) From Eqs. (4.6.2b)/(4.6.5b) and (4.6.3b)/(4.6.6b), we can see that they have linear phase , i.e., their phases are (piecewise) linear in Ω except for phase jumps of ±π or ±2π (see Remark 3.3 and Fig. 4.9(c1) and (c2)).

(3) If a system has the impulse response represented by a symmetric or anti- symmetric sequence of finite duration, such a system has linear phase shifting property so that it will reproduce the input signals falling in the passband with a delay equal to the slope of the phase curve. That is why such a system is called the linear phase FIR filter.

Example 4.12 Pole-zero Pattern of Symmetric or Anti-symmetric Sequences (a) Consider a system whose impulse response is

n=−1 0 1 2 3 4 5

g 1 [n] = [· · · 0 1 −2.5 5.25 −2.5 1.0 0 · · · ]. (E4.12.1) This system has the system function G 1 [z] = Z{g 1 [n]} as

4.6 The z-Transform of Symmetric Sequences 239 Im{z }

Im{z } 4th-order pole at z = 0 3th-order pole at z = 0

4 Re

3 Re { z }

(a1) The pole-zero plot for G 1 [z ]

(a2) The pole-zero plot for G 2 [ z ]

(b1) The magnitude curve G 1 [e jΩ ]

(b2) The magnitude curve G 2 [e jΩ ]

(c2) The phase curve ∠G 2 [ e jΩ ] Fig. 4.9 The pole–zero plots and frequency responses for Example 4.12

(c1) The phase curve ∠G 1 [ e jΩ ]

G 1 [z] = 1 − 2.5z −1 + 5.25z −2 − 2.5z −3 +z −4

j π/ 3 j π/ =z 3 (z − 0.5e )(z − 0.5e − jπ/3 )(z − 2e )(z − 2e − jπ/3 ) (E4.12.2)

whose pole-zero pattern and frequency response magnitude/phase curves are plotted in Fig. 4.9(a1) and (b1)/(c1), respectively.

(b) Consider a system whose impulse response is

n=−1

g 2 [n] = [· · · 0 − 1 2 − 2 1 0 · · · ]. (E4.12.3) This system has the system function G 2 [z] = Z{g 2 [n]} as

2 [z] = −1 + 2z −1 − 2z −2 +z −3 = −z −3 (z − 1)(z − z + 1) j π/ = −z 3 −3 (z − 1)(z − e )(z − e − jπ/3 )

(E4.12.4)

240 4 The z-Transform whose pole-zero pattern and frequency response magnitude/phase curves are

plotted in Fig. 4.9(a2) and (b2)/(c2), respectively. The following program “sig04e12.m” is run to yield Fig. 4.9, where zplane(B,A)

is used to create the pole-zero plot of a system with system function G[z] =

B [z]/ A[z].

%sig04e12.m clear, clf N=360; W=pi/N*[-N:N]; % frequency range for i=1:2

if i==1, B=[1 -2.5 5.25 -2.5 1]; A=1; else B=[-1 2 -2 1]; A=1; %numerator/denominator of system function end figure(1), subplot(220+i), zplane(B,A) GW= freqz(B,A,W); % frequency response GW mag= abs(GW); % magnitude of frequency response GW phase= angle(GW); % phase of frequency response figure(2) subplot(220+i), plot(W,GW mag) set(gca,’fontsize’,9, ’Xlim’,[-pi pi], ’xtick’,[-pi 0 pi], ...

’xticklabel’,{’-pi’ ’0’ ’pi’}) subplot(222+i), plot(W,GW phase) set(gca,’fontsize’,9, ’Xlim’,[-pi pi], ’xtick’,[-pi 0 pi], ...

’xticklabel’,{’-pi’ ’0’ ’pi’}) end