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The unit impulse response of a system is h(t) = e-2t, t 0.
For this system, the steady-state value of the output for unit step input is equal to
Correct answer is '0.5'. Can you explain this answer?
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The unit impulse response of a system is h(t) = e-2t, t 0.For this sys...
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The unit impulse response of a system is h(t) = e-2t, t 0.For this sys...
Explanation:
Unit Impulse Response:
The unit impulse response of a system is the output produced by the system when a unit impulse (delta function) is applied to its input. In this case, the unit impulse response of the system is given by h(t) = e^(-2t), t >= 0.
Steady-State Value:
The steady-state value of the output of a system is the value that the output approaches as time approaches infinity, assuming that the system is stable. In this case, we are given that the input to the system is a unit step function.
Unit Step Function:
A unit step function is a mathematical function defined as u(t) = {0, t < 0;="" 1,="" t="" />= 0}. It is a function that is zero for negative values of t and one for positive values of t.
Output of the System:
The output of the system can be found by convolving the input with the unit impulse response, i.e., y(t) = u(t) * h(t). This can be written as:
y(t) = ∫[0 to t] u(τ)h(t-τ)dτ
Using the given values of the input and the impulse response, we get:
y(t) = ∫[0 to t] e^(-2(t-τ))dτ = 1/2 - 1/2e^(-2t), t >= 0
Steady-State Value of the Output:
As t approaches infinity, the exponential term e^(-2t) approaches zero, and the steady-state value of the output is given by:
y_ss = lim[t→∞] [1/2 - 1/2e^(-2t)] = 1/2
Therefore, the steady-state value of the output for unit step input is 0.5.
Unit Impulse Response:
The unit impulse response of a system is the output produced by the system when a unit impulse (delta function) is applied to its input. In this case, the unit impulse response of the system is given by h(t) = e^(-2t), t >= 0.
Steady-State Value:
The steady-state value of the output of a system is the value that the output approaches as time approaches infinity, assuming that the system is stable. In this case, we are given that the input to the system is a unit step function.
Unit Step Function:
A unit step function is a mathematical function defined as u(t) = {0, t < 0;="" 1,="" t="" />= 0}. It is a function that is zero for negative values of t and one for positive values of t.
Output of the System:
The output of the system can be found by convolving the input with the unit impulse response, i.e., y(t) = u(t) * h(t). This can be written as:
y(t) = ∫[0 to t] u(τ)h(t-τ)dτ
Using the given values of the input and the impulse response, we get:
y(t) = ∫[0 to t] e^(-2(t-τ))dτ = 1/2 - 1/2e^(-2t), t >= 0
Steady-State Value of the Output:
As t approaches infinity, the exponential term e^(-2t) approaches zero, and the steady-state value of the output is given by:
y_ss = lim[t→∞] [1/2 - 1/2e^(-2t)] = 1/2
Therefore, the steady-state value of the output for unit step input is 0.5.
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