some important signal analysis question

1.2 (a) Sketch the signal x(t). Represent the signal in terms of unit step functions.

a) Given x(t)=t+1 when t>=-1 and t<=1

                  And x(t)=0 elsewhere

             At  t=-1   x(t)=0

             At t=0    x(t)=1

             At t=1    x(t)=2                

     x(t)

 

                                                                    -2

                                                                                         -1                                        1 

(b) Sketch the signal s(t)=d/dt.x(t). Represent the signal in terms of unit step functions.

Ans:

s(t)=d/dt x(t)

C)  Find the energy of the signals  x(t),  x(t-1) , s(t)and  s(t-1) . Comment on the results.

Ans:

 1)x(t)=t+1 if    x<=1 and x>=-1

            and 0 elsewhere

       E=   dt

         = dt  +

         =(1/2+1)-(1/2-1)

         =2J

 2)x(t-1)=t if x>0 and x<2

      and 0 if x=2

E=

  =

  =

  =4/2

  =2J

3)s(t)=1 if t<=1 and t>=-1

            And o elsewhere

E=

  =

  =1+1

=2J

4)s(t-1)=1 if t<=2 and t>=0

      And 0 elsewhere

E=

  =

 =2-0

=2J

 From the above four cases we observe that energy does not change for x(t),x(t-1),s(t),s(t-1)

d) Find and sketch the even and odd components of the signals x(t)and s(t). 

Ans:

even signal : (t)=

odd signal: (t)=