PRACTICAL MATLAB® FOR ENGINEERS PRACTICAL MATLAB

Fourier and Laplace 405

axis([-5 5 0 0.6])

subplot(2,2,2);plot(w,phase*180/pi)

xlabel(‘frequency w’);ylabel(‘Angle in degrees’)

title(‘Phase spectrum/Fourier’);

a = [1]; % F(s) = 1/(s+2)

b = [1 2];

H = freqs(a,b,w);

mag = abs(H);ang = angle(H);

degrees = ang*180/pi;

subplot(2,2,3);plot(w,mag)

xlabel(‘frequency s’);ylabel(‘Magnitude of F(s)’)

title(‘Magnitude spectrum/Laplace’);

axis([-5 5 0.1 0.6])

subplot(2,2,4);plot(w,degrees)

xlabel(‘frequency s’);ylabel(‘Angle in degrees’)

title(‘Phase spectrum/Laplace’);

figure(3)

% Fourier transform using the symbolic toolbox

syms v;

disp(‘********************************’)

disp(‘The sym Fourier transform ‘)

disp(‘of(exp(-2*t)*u(t)) is=’)

FT= fourier(exp(-2*v)*sym(‘Heaviside(v)’))

disp(‘********************************’)

% Laplace transform using symbolic toolbox

disp(‘********************************’)

disp(‘The sym Laplace transform ‘)

disp(‘of(exp(-2*t)*u(t)) is=’)

LT=laplace(exp(-2*v)*sym(‘Heaviside(v)’))

disp(‘********************************’)

subplot(2,1,1);

ezplot(abs(FT));

title(‘sym Fourier transf of(exp(-2*t)*u(t))’)

ylabel(‘Magnitude of F(w)’);xlabel(‘frequency w’);

subplot(2,1,2);

ezplot(abs(LT));

title(‘sym Laplace transf of(exp(-2*t)*u(t))’)

ylabel(‘Magnitude of F(s)’);

xlabel(‘frequency s’);axis([-4 0 0 45])

The script fi le Fourier_ Laplace is executed and the results are as follows (Figures 4.52

through 4.54):

>> Fourier _ Laplace

*****************************************

*********** R E S U L T S ************

*****************************************

The int(exp(-2*t)*u(t)) from t=0 to inf is =

1/2

ans =

.50000000000000000000000000000000