Question

7. For each of the following, determine if the following systems are causal and if they are stable. Provide reasons for each answer.

2 a. h(t)= e'u(-1) b. H(s)= ROC -2<Re(s)<-1 (s+1)(s+2) c. H(s)= ROC Re(s)>-1 (s+1)(s+2) d. H(s) = 3 22, Re(s)<2 2

Answer 1

Given h(t) = et ul-t) stable First of all for a system to be two conditions are to be satisfied =) splane 0 The poles should not be present over the right half of s-plane 2 ROC must include jw axis For a system to be causal Sided. ROC must be right

t hits=0 =etult) Apply Caplace transform H(s) - ila S-1 L.T -at e ul-t) a - 1 sta Now Roc is s plane the Re(s)<I → System is not Causal = as pole is on Right half of S-plane System is unstable.

Consider the ROC to splane -2 <Re(s)<-1 - jw axis in not includedin ROC =) Unstable =) Not Right Sided Non-Causal 2 Splane for ROC 7 Rels) @>-| » jw axis is induded Stable → Right Sided causal.

1 (d H(s): S-2 ROC > Re(s) <2 Splanet =) jwaxis not included in ROC unstable 100 2 - Not Not right right sided, Non-causal.