HW 2

%% Homework 2
%% Problem 1
close all
n=[1];
d=conv([1 1],[1 1 20]);
hw=tf(n,d)
t=0:1/1000:10*pi;
x=cos(3.0*t).*heaviside(t)+2*heaviside(t)+cos(10.0*t).*exp(-t).*heaviside(t);
figure
bode(hw)
figure
asymp(hw)
figure
lsim(hw,x,t)

[m3,p3]=bode(hw,3)
[m0,p0]=bode(hw,0)

%% Problem 2
close all
n=[0.1];
d=conv([1 -0.9],[1 -1.8 0.9]);
hw=tf(n,d,1)
t=0:1:300;
x=sin(0.1*t).*heaviside(t+1)+2*heaviside(t);
% figure
% step(hw)
figure
bode(hw)
figure
lsim(hw,x,t)

[mp1,pp1]=bode(hw,0.1)
[m0,p0]=bode(hw,0)

%% Problem 3
close all;
fc=10;
wc=1;
[b,a]=butter(2,wc,'high','s')
hw=tf(b,a)
figure
bode(hw)
[m,p,f]=bode(hw);

freq=0;
for i = 1:1:length(m)
    if m(i) >= 0.99
        freq=f(i);
        break
    end
end


freq
wc=2*pi*fc/freq
[b,a]=butter(2,wc,'high','s')
hw=tf(b,a)
figure
bode(hw,{0.1,1000})
[m,p]=bode(hw,0.1*2*pi);
atten=20*log10(m)

%% Problem 4
close all;
fc=10;
fs=20000;
fn=1000;
Ts=1/fs;
wc=2*pi*fc;

min_wn = 99999;
omega = wc*Ts;
for wn = omega:0.0001:0.05
    m = bode(tf([wn],[1 wn-1],1),omega);
    if m >= 0.99
        if wn < min_wn
            min_wn = wn;
        end
    end
end

wn=min_wn
hw=tf([wn],[1 wn-1],1)
figure
bode(hw)
m = bode(hw,2*pi*fn*Ts)
atten=20*log10(m)