Signals and Systems - Electrical Engineering

180 C H A P T E R 3: The Laplace Transform

FIGURE 3.6

Location of the poles and zeros of

cos( 2 t+θ)u(t)for (a)θ= 0 and for (b)

θ=π/ 4. Note that the zero is moved

to the right to 2 because the zero of

the Laplace transform is

s= 0 tan(θ)=2 tan(π/ 4 )= 2.

− 1 0 1 23

− 2

0

2

jΩ

σ

0 5 10

− 1

−0.5

0

0.5

1

t

x^1

(t

)

jΩ

− 2

0

2

− 1 0123

σ

− 1

−0.5

0

0.5

1

0 5 10

t

x^2

(t

)

(a)

(b)

%%%%%%%%%%%%%%%%%

% Example 3.4

%%%%%%%%%%%%%%%%%

syms t

x = exp (-t);

y = x * cos(10 * t);

X = laplace(x)

Y = laplace(y)

% plotting of signals and poles/zeros

figure(1)

subplot(221)

ezplot(x,[0,5]);grid

axis([0 5 0 1.1]);title(’x(t) = exp(-t)u(t)’)

numx = [0 1];denx = [1 1];

subplot(222)

splane(numx,denx)

subplot(223)

ezplot(y,[-1,5]);grid

axis([0 5 -1.1 1.1]);title(’y(t) = cos(10t)exp(-t)u(t)’)

numy = [0 1 1];deny = [1 2 101];

subplot(224)

splane(numy,deny)

The results are shown in Figure 3.7. n