Consider the linear time invariant system with the impulse response

            .

Determine the output y(t) when the input is

      .


The input and the impulse response of a linear, time-invariant system are shown in Fig. (a) and Fig. (b) respectively.

a)   Determine the output y(t).

b)  Determine if this system is stable. Explain.

 

a)     
Find the impulse response of the causal and LTI system S1 described by:

.

Is this system stable? Explain.

b)      The system in part (a) is cascaded to a system S2 with input-output relationship , as shown in figure (a). Find the response of the cascaded system when the input is a unit impulse.

c)      Repeat part (b) with order of cascade connection is changed as shown in figure (b) and compare the results. Is the output changed? Why?

A linear, time-invariant causal system is characterized by the transfer function   .

Determine and sketch the impulse respone of this system.

The response y(t) of the continuous-time LTI system for the input  is given to be .

a)    Determine the transfer function H(s) of the system.

b)   Determine the impulse response h(t) of the system.


Determine the Laplace transform for the causal, half-wave rectified sine wave shown in the figure.

A linear, time-invariant system is characterized by the differential equation

.

a)   Determine the impulse response of this system.

b)   Determine the output y(t) when the input is x(t)=sin(t).

c)    Is this system stable? Explain.

Determine the impulse response of the LTI system described by the differential equation:

a)  

b)  

c)    .

A signal x(t), which is zero for t negative, has the Laplace transform . Determine:

i)       

ii)     

iii)   


Consider the signal x(t) shown in the figure. Determine the Laplace transform Y(s) of the signal y(t)=x(t)*x(t-1), using the properties of Laplace transforms and well known transform pairs.

A linear, time-invariant causal system is characterized by the differential equation

.

a)   Determine the impulse response of this system.

d)   Is this system stable? Explain.

e)   Making use of the above, determine the impulse response of the LTI system described by the differential equation

.