Problem 8 :
 
 Solution 8 :

  By definition ;

                                       

  Replacing x (t) by cx(t) where c is a real number ,

                              

   Similarly putting ,

                                            , we get

           

                        

                        

  Thus , obeys homogeneity and additivity .

  Implies is LINEAR .


    NEXT , substituting x (t) by x (t-t0)

                                       

 Now , Substituting , ,we get   

                                          

Hence need not be equal to    for all x (t) and t0.

Hence is  shift-variant .


Let       and let x (t) = u (t)

Now ,             

Even though x (t)=0 for  t < 0 ,   is non-zero for t < 0 .

This counterexample shows that the auto correlating system is non-causal.


(b) The system remains linear (homogeneous and additive) and non-causal. The system becomes shift invariant.

                         

  Replacing x (t) by x (t-t0) , we get

                            and

Hence    for all signals x (t) and all values of  t0 .

Thus    is a shift-invariant system .