디지털 신호처리 chapter 8 (Infinite Impulse Response Filter Design) 저자 : Li tan
해당 게시글은 수식표시가 에러로 인해정상표기되고 있지 않은 점 양해 부탁드립니다.
Chapter 8
8.1
8.2
a.
b.
freqz([0.3375 0.3375],[1 -0.3249],512,200)
8.3
a.
b.
>> freqz([0.6225 -0.6225],[1 -0.3594],512,200)
8.4
a.
b =
b.
>> freqz([0.6340 0 0.6340],[1 0 0.2679],512,120)
8.5
a.
b.
>> freqz([0.2113 0 -0.2113],[1 -0.8165 0.5774],512,120)
8.6
a.
b.
>> freqz([0.4005 0.4005],[1 -0.1989],512,8000)
8.7
a.
b.
>> freqz([0.1867 0.3734 0.1867],[1 -0.4629 0.2097],512,8000)
8.8
a.
b.
>> freqz([0.1667 -0.5 0.5 -0.1667],[1 0 0.3333 0],512,8000)
8.9
a.
b.
>> freqz([0.0730 0 -0.0730],[1 0 0.8541],512,8000)
8.10
a.
b.
>> freqz([0.9266 -0.2899 0.9266],[1 -0.2899 0.8532],8000,8000)
8.11
a.
b.
>> freqz([0.5667 0.5667],[1 0.1354],512,8000)
8.12
a.
b.
>> freqz([0.2430 0.4861 0.2430],[1 -0.2457 0.2755],512,8000)
8.13
a.
b.
>> freqz([0.1321 -0.3964 0.3964 -0.1321],[1 0.3432 0.6044 0.2041],512,8000)
8.14
a.
b.
>> freqz([0.1815 0 -0.1815],[1 -0.6265 0.6370],512,8000)
8.15
a.
b.
>> freqz([0.9609 0.7354 0.9609],[1 0.7354 0.9217],512,8000)
8.16
a.
b.
>> freqz([0.0380 0.1519 0.2278 0.1519 0.0380],[1 -0.9784 0.7901 -0.2419 0.0377],512,8000)
8.17
a.
b.
>> freqz([0.0242 0.0968 0.1452 0.0968 0.0242],[1 -1.5895 1.6690 -0.9190 0.2497],512,8000)
8.18
a.
b.
>> freqz([0.00767 0 -0.01534 0 0.00767],[1 -1.4428 2.2935 -1.2918 0.8027],512,8000)
8.19
a.
b. and c.
---------------------------------------------------------------------------------------------------
f=0:0.1:5;T=0.1 % frequency range and sampling interval
w=2*pi*f; % frequency range in radians/sec
hs=freqs([10], [1 10],w); % analog magnitude frequency response
phis=180*angle(hs)/pi;
% for the z-transfer function H(z)
hz=freqz([1],[1 -0.3967],length(w)); % digital magnitude frequency response
phiz=180*angle(hz)/pi;
%plot magnitude and phase responses.
subplot(2,1,1), plot(f,abs(hs),'kx',f, abs(hz),'k-'),grid; axis([0 5 0 2]);
xlabel('Frequency (Hz)'); ylabel('Mag. Responses')
subplot(2,1,2), plot(f,phis,'kx',f, phiz,'k-'); grid;
xlabel('Frequency (Hz)'); ylabel('Phases (deg.)');
-----------------------------------------------------------------------------------------
8.20
a.
b. and c.
----------------------------------------------------------------------------------------------------
f=0:0.1:5;T=0.1; % initialize analog frequency range in Hz and sampling interval
w=2*pi*f; % convert the frequency range to radians/second
hs=freqs([1], [1 3 3],w); % calculate analog filter frequency responses
phis=180*angle(hs)/pi;
% for the z-transfer function H(z)
% calculate digital filter frequency responses
hz=freqz([0.0086],[1 -1.7326 0.7408],length(w));
phiz=180*angle(hz)/pi;
% plot magnitude and phase responses
subplot(2,1,1), plot(f,abs(hs),'kx',f, abs(hz),'k-'),grid;
xlabel('Frequency (Hz)'); ylabel('Magnitude Responses')
subplot(2,1,2), plot(f,phis,'kx',f, phiz,'k-'); grid;
xlabel('Frequency (Hz)'); ylabel('Phases (degrees)')
----------------------------------------------------------------------------------------------------
8.21
a.
b. and c.
---------------------------------------------------------------------------------------------------
f=0:0.1:5;T=0.1; % initialize analog frequency range in Hz and sampling interval
w=2*pi*f; % convert the frequency range to radians/second
hs=freqs([1 0], [1 4 5],w); % calculate analog filter frequency responses
phis=180*angle(hs)/pi;
% for the z-transfer function H(z)
% calculate digital filter frequency responses
hz=freqz([0.1 -0.09781],[1 -1.6293 0.6703],length(w));
phiz=180*angle(hz)/pi;
% plot magnitude and phase responses
subplot(2,1,1), plot(f,abs(hs),'kx',f, abs(hz),'k-'),grid;
xlabel('Frequency (Hz)'); ylabel('Magnitude Responses')
subplot(2,1,2), plot(f,phis,'kx',f, phiz,'k-'); grid;
xlabel('Frequency (Hz)'); ylabel('Phases (degrees)')
-----------------------------------------------------------------------------------------------------------------------
8.22
8.23
8.24
8.25
8.26
8.27
8.28
a.
b.
Direct-form I:
Direct-form II:
8.29
a.
b. for section 1: 
for section 2: 
8.30
Chebyshev notch filter: order =2
-------------------------------------------------------------------------------------
fs=8000;T=1/fs;
w0=2*pi*360; wa0=(2/T)*tan(w0*T/2);
wL=2*pi*330; waL=(2/T)*tan(wL*T/2);
wH=2*pi*390; waH=(2/T)*tan(wH*T/2);
waaL=wa0*wa0/waH;BW1=waH-waaL
waaH=wa0*wa0/waL;BW2=waaH-waL
[B,A]=lp2bs(2.8628,[1 2.8628],wa0,BW2);
[b,a]=bilinear(B,A,fs)
freqz(b,a,8000,8000);
-------------------------------------------------------------------------------
8.31 Chebyshev notch filter 1: order =2
See Problem 8.31.
Chebyshev notch filter 2: order =2
------------------------------------------------------------------------------------
fs=8000;T=1/fs;
w0=2*pi*1080; wa0=(2/T)*tan(w0*T/2);
wL=2*pi*1050; waL=(2/T)*tan(wL*T/2);
wH=2*pi*1110; waH=(2/T)*tan(wH*T/2);
waaL=wa0*wa0/waH;BW1=waH-waaL
waaH=wa0*wa0/waL;BW2=waaH-waL
[B,A]=lp2bs(2.8628,[1 2.8628],wa0,BW2);
[b,a]=bilinear(B,A,fs)
freqz(b,a,8000,8000);
------------------------------------------------------------------------------------
8.32
radians/se.
radians/sec
and 
dB
Choose 
Butterworth filter order = n=4
-------------------------------------------------------------------------------------------
fs=10000;T=1/fs;
wd=2*pi*3000; wa=(2/T)*tan(wd*T/2);
[B,A]=lp2lp(1,[ 1 2.6131 3.4142 2.6131 1],wa);
[b,a]=bilinear(B,A,fs)
freqz(b,a,512,fs);
-----------------------------------------------------------------------------------------
8.33
radians/second,
radians/second, and 
sec.
radians/se.
radians/sec
and 
dB
Chebyshev filter order = 4;
---------------------------------------------------------------------------------------------------
fs=10000;T=1/fs;
wd=2*pi*3000; wa=(2/T)*tan(wd*T/2);
[B,A]=lp2lp(0.2456,[ 1 0.9528 1.4539 0.7426 0.2756],wa);
[b,a]=bilinear(B,A,fs)
freqz(b,a,512,fs);
-----------------------------------------------------------------------------------------------------------------------
8.34 
and 
>> r=1-500*pi/8000;
>> theta=1750*2*pi/8000;
>> K=(1-r)*sqrt(1-2*r*cos(2*theta)+r*r)/(2*abs(sin(theta)));
r = 0.8037
K =0.1771
-2*r*cos(theta)
ans = -0.3136
8.35
Fixing 
Fixing 
,
, 
We choose a smaller bandwidth for aggressive design:
,
and 
dB
Choose 
filter order 2n= 4
------------------------------------------------------------------------------------------------------------------------
fs=8000;T=1/fs;
wL=2*pi*1500; waL=(2/T)*tan(wL*T/2);
wH=2*pi*2000; waH=(2/T)*tan(wH*T/2);
wa0=sqrt(waL*waH); BW=waH-waL
[B,A]=lp2bp(1,[1 1.4142 1],wa0,BW);
[b,a]=bilinear(B,A,fs)
freqz(b,a,512,8000);
--------------------------------------------------------------------------------------------------------------------------
8.36
a.
b.
c.
---------------------------------------------------------------------------------------------------
fs=44100;T=1/fs;
wd=2*pi*1000; wa=(2/T)*tan(wd*T/2);
[B,A]=lp2lp(1,[ 1 2 2 1],wa);
[bL,aL]=bilinear(B,A,fs)
freqz(bL,aL,512,fs);
[hL,ff]=freqz(bL,aL,512,fs);
figure
[B,A]=lp2hp(1,[ 1 2 2 1],wa);
[bH,aH]=bilinear(B,A,fs)
freqz(bH,aH,512,fs);
[hH,ff]=freqz(bH,aH,512,fs);
figure
H=abs(hL)+abs(hH);
plot(ff,20*log10(abs(hL)),ff,20*log10(abs(hH)),'-.', ff,20*log10(H));
-----------------------------------------------------------------------------------------------------
8.37
a.
b.
c.
>> fs=8000; T=1/fs;
>> t=0:T:0.01;
>> x=zeros(1,length(t)); x(1)=1;
>> y=filter([0 0.5878],[1 -1.6180 1],x);
>> plot(t,y);grid;xlabel('Time (sec.)');ylabel('800-Hz tone');
8.38
a.
b.
c.
-------------------------------------------------------------------------------------------
fs=8000;T=1/fs;
x=zeros(1,205);x(1)=1;
y1=filter([0 sin(2*pi*770/fs)],[1 -2*cos(2*pi*770/fs) 1],x);
y2=filter([0 sin(2*pi*1336/fs) ],[1 -2*cos(2*pi*1336/fs) 1],x);
y=y1+y2;
plot(N,y,'k');grid
ylabel('y(n) DTMF: number 5');xlabel('Sample number n')
------------------------------------------------------------------------------------------
8.39
for 
with 
, 
a. 
; b. 
; c. 
for
with 
, 
d. 
; e. 
; f. 
8.40
, and 
a.
and 
b.
with 
, for 
with , for
c.
and
d.
--------------------------------------------------------------------------------------------
fs=8000;T=1/fs;
x=zeros(1,205);x(1)=1;
y1=filter([0 sin(2*pi*770/fs)],[1 -2*cos(2*pi*770/fs) 1],x);
y2=filter([0 sin(2*pi*1336/fs) ],[1 -2*cos(2*pi*1336/fs) 1],x);
y=y1+y2;
xDTMF=[y 0];
v20=filter(1,[1 -2*cos(2*pi*20/205) 1],xDTMF);
v34=filter(1,[1 -2*cos(2*pi*34/205) 1],xDTMF);
X20=sqrt(v20(206)^2+v20(205)^2-2*cos(2*pi*20/205)*v20(206)*v20(205));
X34=sqrt(v34(206)^2+v34(205)^2-2*cos(2*pi*34/205)*v34(206)*v34(205));
A20=2*X20/205
A34=2*X34/205
-----------------------------------------------------------------------------------------------
A20 = 0.8818
A34 = 0.9147
8.41
a.
--------------------------------------------------------------------------------------------------
n=0:N-1;
sample=1.2*sin(2*pi*1000*n/10000)-1.5*cos(2*pi*4000*n/10000);
%direct-form I implementation
x=[0 0 0 0 0]; %input buffer [x(n) x(n-1) ..]
y=[0 0 0 0 0]; %output buffer [y(n) y(n-1) ... ]
b=[0.1103 0.4412 0.6618 0.4412 0.1603]; %Numerator coefficients [b0 b1 ...]
a=[1 0.1509 0.9041 -0.1619 0.1872]; %Denominator coefficients [1 a1 ...]
KKb=length(b); KKa=length(a);
for n=1:1:length(sample) % loop processing
for k=KKb:-1:2 % shift input by one sample
x(k)=x(k-1);
end
x(1)=sample(n); % get new sample
for k=KKa:-1:2 % shift input by one sample
y(k)=y(k-1);
end
y(1)=0; % perform IIR filtering
for k=1:1:KKb
y(1)=y(1)+b(k)*x(k);
end
for k=2:1:KKa
y(1)=y(1)-a(k)*y(k);
end
out(n)=y(1); %send filtered sample to the output array
end
subplot(2,1,1);plot(sample);grid;axis([0 500 -3 3]);
xlabel('Smaple number n');ylabel('Sample(n)');
subplot(2,1,2);plot(out);grid;axis([0 500 -3 3]);
xlabel('Smaple number n');ylabel('Out(n)');
-------------------------------------------------------------------------------------------
b.
-----------------------------------------------------------------------------------------------
n=0:N-1;
sample=1.2*sin(2*pi*1000*n/10000)-1.5*cos(2*pi*4000*n/10000);
%direct-form II implementation
w=[0 0 0 0 0]; %filter states [w(n) w(n-1) ..]
b=[0.1103 0.4412 0.6618 0.4412 0.1603]; %Numerator coefficients [b0 b1 ...]
a=[1 0.1509 0.9041 -0.1619 0.1872]; %Denominator coefficients [1 a1 ...]
KKb=length(b); KKa=length(a); KKw=length(w);
for n=1:1:length(sample) % loop processing
for k=KKw:-1:2 % shift input by one sample
w(k)=w(k-1);
end
w(1)=sample(n);
for k=2:1:KKa
w(1)=w(1)-a(k)*w(k); %IIR filtering
end
sum=0;
for k=1:1:KKb
sum=sum+b(k)*w(k); %FIR filtering
end
out(n)=sum; %send filtered sample to the output array
end
subplot(2,1,1);plot(sample);grid;axis([0 500 -3 3]);
xlabel('Smaple number n');ylabel('Sample(n)');
subplot(2,1,2);plot(out');grid;axis([0 500 -3 3]);
xlabel('Smaple number n');ylabel('Out(n)');
-------------------------------------------------------------
Plots are the same as ones in (b).
'공부하자 > 아무거나 공부하자' 카테고리의 다른 글
| 2017학년도 일본정부 장학금 유학생 모집공문 (0) | 2016.12.12 |
|---|---|
| 네이버 웨일브라우저 (1차,2차 테스트 추가 신청) (0) | 2016.12.12 |
| 던파 스위칭의 모든것 (0) | 2016.12.06 |
| 던파 루크레이드 공대로 보는 직업순위 및 통계(출처 던파인벤) (0) | 2016.12.06 |
| 데이터스튜디오 실험1(dataStudio) 물리실험 (솔루션 파일) (0) | 2016.12.05 |
