Recall how to determine the impulse response of a system by substituti

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Recall how to determine the impulse response of a system by substituting x(t) = δ(t) and y(t) = T[δ(t)] = h(t). Now, for each of the following LTI system T[ ], determine the impulse response. [5×4 = 20] (a) T[x(t)] = −12 x(t−1)+x(t)+12 x(t+1) (b) T[x(t)] = etx(t−1) (c) T[x(t)] = ∫−12 x(t+1−τ)dτ (d) T[x(t)] = ∫−∞tsin⁡(t−τ)x(τ)dτ

Recall how to determine the impulse response of a system by substituting x(t) = δ(t) and y(t) = T[δ(t)] = h(t). Now, for each of the following LTI system T[ ], determine the impulse response. [5×4 = 20] (a) T[x(t)] = −12 x(t−1)+x(t)+12 x(t+1) (b) T[x(t)] = etx(t−1) (c) T[x(t)] = ∫−12 x(t+1−τ)dτ (d) T[x(t)] = ∫−∞tsin⁡(t−τ)x(τ)dτ

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Recall how to determine the impulse response of a system by substituting

x (t) = δ (t)

and

y (t) = T [δ (t)] = h (t)

. Now, for each of the following LTI system

T [

], determine the impulse response.

[5 \times 4 = 20]

(a)

T [x (t)] = - \frac{1}{2} x (t - 1) + x (t) + \frac{1}{2} x (t + 1)

(b)

T [x (t)] = e^{t} x (t - 1)

(c)

T [x (t)] = \int_{- 1}^{2} x (t + 1 - τ) d τ

(d)

T [x (t)] = \int_{- \infty}^{t} \sin (t - τ) x (τ) d τ

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