Problem 2 Draw a direct-form representation for the causal Lti system with system H(s) s(s+ (s+3)(s+4) Problem 3 Consider the cascade of two LTI systems as depicted below () (t) System A System B where we have the following: System A is causal with impulse response System B is causal and is characterized by the following differential equation relating its input, w(t), and output, y(t) dy (t) dt+y(t) d() If the input c(t) output y(t)=0 (a) Find the system function H(s)=Y(s/X(s), determine its ROC and sketch its pole- zero pattern. Note: Your answer should only have numbers in them (i.e, you have enough information to determine the value of a) (b)Determine the differential equation relating y(t)and a(t) Problem 4 Suppose we are given the following information about a causal and stable LtI system with impulse response h(t)and a rational function H(s) The steady state response to a unit step, i.e., s(oo)= When the input is eu(t), the output is absolutely integrable · The signal h(t).dh(t) dt2 +6h(t) dt is of finite duration h(t) has exactly one zero at infinity Determine H(s) and its ROCProblem 2 Draw a direct-form representation for the causal LTI system with system function H(s) = s(s + 1) (s + 3)(s + 4). Problem 3 Consider the cascade of two LTI systems as depicted below: x(t) w(t) System A System B y(t) where we have the following: • System A is causal with impulse response h(t) = e−2t u(t) • System B is causal and is characterized by the following differential equation relating its input, w(t), and output, y(t): dy(t) dw(t) + y(t) = + �w(t) dt dt • If the input x(t) = e−3t , the output y(t) = 0. (a) Find the system function H(s) = Y (s)/X(s), determine its ROC and sketch its pole￾zero pattern. Note: Your answer should only have numbers in them (i.e., you have enough information to determine the value of �). (b) Determine the differential equation relating y(t) and x(t). Problem 4 Suppose we are given the following information about a causal and stable LTI system with impulse response h(t) and a rational function H(s): • s(� 1 The steady state response to a unit step, i.e., ) = . 3 • When the input is et u(t), the output is absolutely integrable. • The signal d2h(t) dh(t) + 5 + 6h(t) dt2 dt is of finite duration. • h(t) has exactly one zero at infinity. Determine H(s) and its ROC. 2