Question

Two LTI systems are connected in series. Their impulse responses are

h1(t)=e⁻³ᵗu(t),h2(t)=10e⁻⁵ᵗu(t)
(a)Prove that the impulse response h(t) of the composite system equals the convolution of h3(t) and h2(t).

Answer 1

The impulse response h(t) of the composite system equals the convolution of h1(t) and h2(t). Using the convolution integral, we find h(t) = (1/8)e^(-5t)(1 - e^(-8t)).

Step-by-step explanation:

To prove that the impulse response h(t) of the composite system equals the convolution of h3(t) and h2(t), we need to find the convolution of h1(t) and h2(t). The convolution of two functions f(t) and g(t) is given by the integral of f(tau)g(t-tau) with respect to tau. In this case, h(t) = h1(t) * h2(t) = ∫e^(-3tau)10e^(-5(t-tau))u(t-tau)dtau, which simplifies to h(t) = 10∫e^(-8tau)e^(-5t)u(t-tau)dtau. Using the property of the unit step function, we can simplify further to h(t) = 10e^(-5t)∫e^(-8tau)dtau. Integrating the exponential function, we get h(t) = 10e^(-5t)(-1/8)e^(-8tau) from 0 to t, which simplifies to h(t) = (1/8)e^(-5t)(1 - e^(-8t)).