1. A linear, time invariant system has an impulse response h(t)=u(t). Determine the output of the system with the input u(t)=µ(t). ( µ (t) = ). [1; t≥ 0 0;t<0².
Please show solution on how to get the answer. Thank you
Introduction :
The answer to this question can be determined by using the convolution integral. The convolution integral is a mathematical operation that is used to determine the output of a system given an input signal and the system's impulse response.
The convolution integral equation is given by:
y(t) = ∫ h(τ)u(t-τ) dτ
For this problem, the impulse response is h(t)=µ(t), and the input signal is u(t)=µ(t). Substituting these values into the convolution integral equation yields:
y(t) = ∫ µ(τ)µ(t-τ) dτ
Since the impulse response and input are both equal to a unit-step function, the convolution integral simplifies to:
y(t) = ∫ µ(t-τ) dτ
Integrating both sides yields:
y(t) = t*µ(t)
Therefore, the output of the system with the given input is y(t)=t*µ(t).
- Algorithm:
1. Initialize the output of the system y(t) to zero
2. Iterate through each value of t
3. For each value of t, if t is greater than or equal to 0, evaluate y(t) as y(t) = h(t)*u(t)
4. At the end, the output y(t) will be equal to µ(t)
- linearity:
linearity, time invariance, and impulse response. Linearity refers to the fact that in an LTI system, the output is always linearly related to the input. Time invariance means that the system's output is the same for any given input, regardless of the time it is applied. The impulse response of a system is the system's output when given an impulse input.
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