update

% 4.2(a)

% 4.2(b)

tau = 0.01

T = 10

N = T/tau

tau=0.01;

T=10;

N=T/tau;

t = [0:tau:T-tau]

y = exp(-2 * abs(t-5))

t=(0:tau:T-tau);

y=exp(-2 * abs(t-5));

%4.2(c)

Y = fftshift(tau*fft(y))

Y = fftshift(tau*fft(y));

%4.2(d

w = -(pi/tau)+(0:N-1)*(2*pi/(N*tau))

w = -(pi/tau)+(0:N-1)*(2*pi/(N*tau));

%4.2(e)

X = exp(j*5*w).*Y

X = exp(1i*5*w).*Y;

%4.2(f)

magnitudeX_app = abs(X)

phaseX_app = angle(X)

magnitudeX_app = abs(X);

phaseX_app = angle(X);

x = exp(-2 * abs(t))

X2 = fftshift(tau * fft(x))

magnitudeX = abs(X2)

phaseX = angle(X2)

X2 = 1 ./ (2 + 1j * w) + 1 ./ (2 - 1j * w);

magnitudeX = abs(X2);

phaseX = angle(X2);

figure;

subplot(2, 1, 1)

hold on

semilogy(w, magnitudeX_app , w, magnitudeX)

subplot(2, 1, 1);

semilogy(w, magnitudeX_app , w, magnitudeX);

legend('|X| by approximation', '|X| ', 'Location', 'northeast');

title('magnitude of X and approximation of X');

subplot(2, 1, 2)

hold on

semilogy(w, phaseX_app, w, phaseX)

subplot(2, 1, 2);

hold on;

semilogy(w, phaseX_app, w, phaseX);

legend('|X| by approximation', '|X|', 'Location', 'northeast');

title('phase of X and approximation of X');

saveas(gcf, "plots/P4_2_f_out1.png")

close

close;

%4.2(g)

magnitudeY = abs(Y)

phaseY = angle(Y)

magnitudeY = abs(Y);

phaseY = angle(Y);

figure;

subplot(2, 1, 1)

hold on

semilogy(w, magnitudeY , 'g')

semilogy(w, magnitudeX, 'b')

subplot(2, 1, 1);

semilogy(w, magnitudeY , 'g--h',w, magnitudeX_app, 'b');

legend('Y', 'X', 'Location', 'northeast');

title('magnitude of X and Y');

subplot(2, 1, 2)

hold on

semilogy(w, phaseY, 'g')

semilogy(w, phaseX, 'b')

subplot(2, 1, 2);

hold on;

semilogy(w, phaseY, 'g');

semilogy(w, phaseX, 'b');

legend('Y', 'X', 'Location', 'northeast');

title('phase of X and Y');

saveas(gcf, "plots/P4_2_f_out2.png")

close

% 4.5(a)

a1 = [1 1.5 0.5]

b1 = [1 -2]

a1 = [1 1.5 0.5];

b1 = [1 -2];

% 4.5(b)

[r1, p1] = residue(b1, a1)

[r1, p1] = residue(b1, a1);

% r1 = [6, -5]

% p1 = [-1, -0.5]

% 4.5(c)

% 4.5(d)

a2 = [1 7 16 12]

b2 = [3 10 5]

a2 = [1 7 16 12];

b2 = [3 10 5];

[r2, p2] = residue(b2, a2)

[r2, p2] = residue(b2, a2);

% r2 = [2 1 -3]

% p2 = [-3 -2 -2]

% 4.5(g)

a3 = [1 -4]

b3 = [-4]

a3 = [1 0 -4];

b3 = [-4];

% 4.5(h)

[r3, p3] = residue(b3, a3)

[r3, p3] = residue(b3, a3);

% r3 = [-4]

% p3 = [4]

% r3 = [-1 1]

% p3 = [2 -2]

% 4.5(i)

load ctftmod.mat

% 4.6(a)

z = [dash dash dot dot]

z = [dash dash dot dot];

% 4.6(b)

figure;