Signals and Systems - Electrical Engineering

Problems 227

and zero elsewhere. The raised cosine is used in communications to transmit signals with minimal

interference. Find its Laplace transform and its corresponding region of convergence.

(c) Indicate three possible approaches to finding the Laplace transform ofcos^2 (t)u(t). Use two of these

approaches to find the Laplace transform.

3.4. Unit-step signals and the Laplace transform
Find the Laplace transform of the reflection of the unit-step signal (i.e.,u(−t)) and its region of conver-
gence. Then use the result together with the Laplace transform ofu(t)to see if you can obtain the Laplace
transform of a constant oru(t)+u(−t)(assumeu( 0 )=0.5so there is no discontinuity att= 0 ).

3.5. Laplace transform of noncausal signal
Consider the noncausal signal

x(t)=e−|t|u(t+ 1 )

Carefully plot it, and find its Laplace transformX(s)by separatingx(t)into a causal signal and an anti-

causal signal,xc(t)andxac(t), respectively, and plot them separately. Find the ROC ofX(s),Xc(s), and

Xac(s).

3.6. Transfer function and differential equation
The transfer function of a causal LTI system is

H(s)=

1

s^2 + 4

(a) Find the differential equation that relates the inputx(t)and the outputy(t)of the system.

(b) Suppose we would like the outputy(t)to be identically zero fortgreater or equal to zero. If we let

x(t)=δ(t), what would the initial conditions be equal to?

3.7. Transfer function
The input to an LTI system is

x(t)=u(t)− 2 u(t− 1 )+u(t− 2 )

If the Laplace transform of the output is given by

Y(s)=

(s+ 2 )( 1 −e−s)^2 )

s^2 (s+ 1 )^2

determine the transfer function of the system.

3.8. Inverse Laplace transform—MATLAB
Consider the following inverse Laplace transform problems for a causal signalx(t):
(a) Given the Laplace transform

X(s)=

s^4 + 2 s+ 1

s^3 + 4 s^2 + 5 s+ 2

which is not proper, determine the amplitude of theδ(t)anddδ(t)/dtterms in the inverse signalx(t).

(b) Find the inverse Laplace transform of

X(s)=

s^2 − 3

(s+ 1 )(s+ 2 )

Can you use the initial-value theorem to check your result? Explain.