Question

Consider an LTI system with input and output related through the equation

y(t)=∫ᵗ₋ₓ e⁻(⁻ᵗ⁻ᵀ) x(τ−2)dτ

What is the impulse response h(t) for this system?

Answer 1

The impulse response for this system is h(t) = e⁻(⁻ᵗ⁻ᵀ).

Step-by-step explanation:

To find the impulse response, we need to input an impulse signal into the system and observe the output. An impulse signal is a signal with a very short duration and a very large magnitude. In this case, the input signal x(t) would be a Dirac delta function, denoted as δ(t).

Substituting this input signal into the equation y(t)=∫ᵗ₋ₓ e⁻(⁻ᵗ⁻ᵀ) x(τ−2)dτ, we get y(t)=∫ᵗ₋ₓ e⁻(⁻ᵗ⁻ᵀ) δ(τ−2)dτ. Using the sifting property of the Dirac delta function, we can simplify this to y(t) = e⁻(⁻ᵗ⁻ᵀ).

Therefore, the impulse response h(t) for this system is h(t) = e⁻(⁻ᵗ⁻ᵀ).

Answer 2

When the input is an impulse function, the output is equal to the impulse response of the LTI system. Therefore, the impulse response (h(t)) for the provided LTI system is h(t) = e⁻(⁻ᵗ⁻ᵀ) δ(t−2).

Step-by-step explanation:

The provided equation represents an LTI (Linear Time-Invariant) system, where the input and output are related through the equation y(t) = ∫ᵗ₋ₓ e⁻(⁻ᵗ⁻ᵀ) x(τ−2)dτ. To determine the impulse response h(t) for this system, we need to find the output when the input is an impulse function, δ(t).

When an impulse function is used as the input, the output is equal to the system's impulse response. Therefore, to find h(t), we substitute δ(t) into the equation:

y(t) = ∫ᵗ₋ₓ e⁻(⁻ᵗ⁻ᵀ) δ(τ−2)dτ

Since the integral of an impulse function (δ(t)) is equal to 1, the equation simplifies to:

y(t) = e⁻(⁻ᵗ⁻ᵀ) δ(t−2)

Therefore, the impulse response h(t) for the given LTI system is h(t) = e⁻(⁻ᵗ⁻ᵀ) δ(t−2).