sample project questions on filter

1. w_w0=0:0.02:3;
Aw2k1=sqrt(1./(w_w0.^2+1));
Aw2k2=sqrt(1./(w_w0.^4+1));
Aw2k4=sqrt(1./(w_w0.^8+1));
Aw2k8=sqrt(1./(w_w0.^16+1));
plot(w_w0,Aw2k1,w_w0,Aw2k2,w_w0,Aw2k4,w_w0,Aw2k8);
Grid
f=1:10:5000;
R1=11300;
R2=22600; R3=68100;
R4=R3;
C1=10^(-8);
C2=C1;
a=sqrt(2); b=1; w=2*pi*f; fc=sqrt(1/(b*R1*R2*C1*C2))/(2*pi);
wc=2*pi*fc;
K=1+R3/R4; s=w*j; Gw=(K.*b.*wc.^2)./(s.^2+a.*wc.*s+b.*wc.^2);
magGw=abs(Gw);
semilogx(f,magGw);
grid;
hold on;
xlabel ('Frequency Hz'); ylabel('|Vout/Vin|');
title ('2nd Order Butterworth Low-Pass Filter Response')

2. a=sqrt(2); b =1; C1=10^(-8); C2=C1; fc=1000; wc=2*pi*fc; K=2;
R2=(4*b)/(C1*sqrt(a^2+8*b*(K-1))*wc);
R1=b/(C1^2*R2*wc^2); R3=(K*R2)/(K-1); R4=K*R2; fprintf(' n');
fprintf('R1 = %5.0f Ohms t',R1); fprintf('R2 = %5.0f Ohms t',R2);
fprintf('R3 = %5.0f Ohms t',R3); fprintf('R4 = %5.0f Ohms t',R4)
f=10:10:20000; w=2*pi*f; R1=12700; R2=20000; R3=40200; R4=R3; K=1+R4/R3;
wc=(4*b)/(C1*sqrt(a^2+8*b*(K-1))*R2);
s=w*j;
Gw=(K.*s.^2)./(s.^2+a.*wc.*s./b+wc.^2./b);
semilogx(f,abs(Gw)); grid; hold on;
xlabel('Frequency, Hz'), ylabel('|Vout/Vin|');
title('2nd Order Butterworth High-Pass Filter Response')
fplot('cos(x)',[-2*pi 2*pi -1.2 1.2])
fplot('sin(x)./x',[-20 20 -0.4 1.2])
fplot(@(x)cos(x)), [-2*pi 2*pi -1.2 1.2]
fplot(@(x)sin(x)./x)), [-20 20 -0.4 1.2]
Wp=10; Ws=16.5; r=2;Gs=-20;
[n,Wp]=chebylord (Wp,Ws,r,-Gs, 's');
[num, den]=chebyl (n ,r, Wp, 's' );

3. %%%%%%%%%%%%%%%%%%%
% Example 6.8 -- Filtering with Butterworth filter
%%%%%%%%%%%%%%%%%%%
clear all; clf
syms t w
x = cos(10 * t) - 2 * cos(5 * t) + 4 * sin(20 * t);% input signal
X = fourier(x);
N = 3; Whp = 5; % filter parameters
[b, a] = butter(N, Whp, ’s’); % filter design
W = 0:0.01:30; Hm = abs(freqs(b, a, W)); % magnitude responsein W
% filter output
n = N:-1:0; U = ( j * w).ˆn
num = b - conj(U’); den = a - conj(U’);H = num/den; % frequency response
Y = X * H; % convolution property
y = ifourier(Y, t); % inverse Fourier
%%%%%%%%%%%%%%%%%%%
% Example 6.9 -- Filtering with Butterworth and Chebyshev filters
%%%%%%%%%%%%%%%%%%%
clear all;clf
syms t w
x = cos(10 ∗ t) - 2 ∗ cos(5 ∗ t) + 4 ∗ sin(20 ∗ t); X = fourier(x)
wp = 5;ws = 10;alphamax = 0.1;alphamin = 15; % filter parameters
% butterworth filter
[N, whp] = buttord(wp, ws, alphamax, alphamin, ’s’)
[b, a] = butter(N, whp, ’s’)
% cheby1 filter
epsi= sqrt(10ˆ(alphamax/10) -1)
wp = whp/cosh(acosh(1/epsi)/N) % recomputing wp to get same whp
[N1, wn] = cheb1ord(wp, ws, alphamax, alphamin, ’s’);
[b1, a1] = cheby1(N1, alphamax, wn, ’s’);
% frequency responses
W = 0:0.01:30;
Hm = abs(freqs(b, a, W));
Hm1 = abs(freqs(b1, a1, W));
% generation of frequency responsefrom coefficients
n = N:-1:0; n1 = N1:-1:0;
U = ( j * w).^n; U1 = ( j *w).^n1
num = b .*conj(U’); den = a .* conj(U’);
num1 = b1 conj(U1’); den1 = a1 ∗ conj(U1’)
H = num/den; % Butterworth LPF
H1 = num1/den1; % Chebyshev LPF
% output of filter
Y = X ∗ H;
Y1 = X ∗ H1;
y = ifourier(Y, t)
y1 = ifourier(Y1, t