Question

The impulse response of an LTI system is given by h(t)=e⁻⁹ᵗu(t) Compute the output y(t) for each input x(t).

x(t)=e⁻⁹ᵗu(t)
y(t)=___

Answer 1

The output y(t) for the given input x(t) can be calculated by convolving the input signal with the impulse response. The resulting equation for y(t) is -1/9e^(-9t) + 1/9.

Step-by-step explanation:

To calculate the output y(t), we need to convolve the input signal x(t) with the impulse response h(t). The convolution operation is represented by the integral of the product of the input signal and the impulse response. In this case, the convolution integral would be:

y(t) = ∫[x(τ)⋅h(t-τ)]dτ

Plugging in the given values, we have:

y(t) = ∫[e^(-9τ)⋅e^(-9(t-τ))]dτ

Simplifying the integrand, we get:

y(t) = ∫e^(-9t)dτ

Integrating with respect to τ, we get:

y(t) = -1/9e^(-9t) + C

Since the system is linear and time-invariant, the constant C can be determined by the initial conditions. If we assume that the system is at rest prior to t=0, then the output at t=0 would be zero. Therefore, C = 1/9. Substituting this value back into the equation, we have:

y(t) = -1/9e^(-9t) + 1/9