Signals and Systems - Electrical Engineering

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

(a) Use the integral formula to findP(s), the Laplace transform of p(t). Determine the region of

convergence ofP(s).

(b) Representp(t)in terms of the unit-step function and use its Laplace transform and the time-shift

property to findP(s). Find the poles and zeros ofP(s)to verify the region of convergence obtained

above.

3.15. Frequency-shift property

Duality occurs between time and frequency shifts. As shown, ifL[x(t)]=X(s), thenL[x(t−t 0 )]=

X(s)e−t^0 s. The dual of this would beL[x(t)e−αt]=X(s+α), which we call thefrequency-shiftproperty.

(a) Use the integral formula for the Laplace transform to show the frequency-shift property.

(b) Use the above frequency-shift property to findX(s)=L[x(t)=cos( 0 t)u(t)](represent the cosine

using Euler’s identity). Find and plot the poles and zeros ofX(s).

(c) Recall the definition of the hyperbolic cosine,cosh( 0 t)=0.5(e^0 t+e−^0 t), and find the Laplace

transformY(s)ofy(t)=cosh( 0 t)u(t). Find and plot the poles and zeros ofY(s). Explain the relation

of the poles ofX(s)andY(s)by connectingx(t)withy(t).

3.16. Poles and zeros

Consider the pulsex(t)=u(t)−u(t− 1 ).

(a) Find the zeros and poles ofX(s)and plot them.

(b) Supposex(t)is the input of an LTI system with a transfer functionH(s)= 1 /(s^2 + 4 π^2 ). Find and plot

the poles and zeros ofY(s)=L[y(t)]=H(s)X(s)wherey(t)is the output of the system.

(c) If the transfer function of the LTI system is

G(s)=

Z(s)

X(s)

=

∏∞

k= 1

1

s^2 +( 2 kπ)^2

and the input is the above signalx(t), compute the outputz(t).

3.17. Poles and zeros—MATLAB

The poles corresponding to the Laplace transformX(s)of a signalx(t)are

p1,2=− 3 ±jπ/ 2

p 3 = 0

(a) Within some constants, give a general form of the signalx(t).

(b) Let

X(s)=

1

(s+ 3 −jπ/ 2 )(s+ 3 −jπ/ 2 )s

From the location of the poles, obtain a general form forx(t). Use MATLAB to findx(t)and plot it. How

well did you guess the answer?

3.18. Solving differential equations—MATLAB

One of the uses of the Laplace transform is the solution of differential equations.

(a) Suppose you are given the differential equation that represents an LTI system,

y(^2 )(t)+0.5y(^1 )(t)+0.15y(t)=x(t) t≥ 0

wherey(t)is the output andx(t)is the input of the system, andy(^1 )(t)andy(^2 )(t)are first- and second-

order derivatives with respect tot. The input is causal, (i.e.,x(t)= 0 t< 0 ). What should the initial

conditions be for the system to be LTI? FindY(s)for those initial conditions.