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 .

 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 .