Math Problem Statement
An analog LTI system with impulse response function h(t) = u(3 − t) + u(t − 3) is
presented with an input x(t) = (t-1)[u(t − 1) − u(t − 5)].
a) Determine and plot the system output y(t) = x(t) ∗ h(t)
Solution
The plots show the input signal , the impulse response , and the output signal .
- Input Signal : A ramp that starts at and stops at .
- Impulse Response : Defined by two shifted step functions centered at , creating a window between and .
- Output Signal : This is the result of convolving and . The output represents the combined effect of the input signal passing through the system characterized by .
Would you like more details or have any questions?
Here are some related questions for deeper exploration:
- How would the output change if were a different step function?
- What happens if the input signal is extended beyond ?
- Can we derive an analytical expression for ?
- How does the duration of the impulse response affect the output?
- What is the physical interpretation of the convolution result in this system?
Tip: When convolving signals, always consider the duration and overlap of the signals to predict the result's shape.
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Math Problem Analysis
Mathematical Concepts
Convolution
Linear Time-Invariant (LTI) Systems
Step Functions
Signal Processing
Formulas
Convolution: y(t) = (x * h)(t) = ∫ x(τ)h(t − τ) dτ
Unit Step Function: u(t)
Theorems
Convolution Theorem
Suitable Grade Level
Undergraduate (Engineering/Applied Mathematics)
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Convolution of Signals: x(t) = u(t) - 2u(t - 1) + u(t - 2) and h(t) = u(t + 1) - u(t - 1)