Math Problem Statement
Solution
To find the maximum value of amplitude at the output of the system, let’s analyze the given information:
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Input Signal :
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Impulse Response : where is the unit step function, implying that for .
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Output : Since this is a Linear Time-Invariant (LTI) system, the output is given by the convolution of and :
To calculate , let’s examine the convolution:
Step-by-Step Solution:
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The Fourier Transform of is:
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The Fourier Transform of is:
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The output in the frequency domain is .
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Calculating at (the frequency component of ): The magnitude of this term is .
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Since the amplitude of is scaled by , the maximum amplitude of the output will be:
Answer:
The maximum amplitude of the output is .
Would you like further clarification on any part of this solution?
Here are some related questions for further exploration:
- What is the significance of convolution in LTI systems?
- How does the frequency response of a system affect the output amplitude?
- Can this method be used for any input signal? Why or why not?
- How would the output change if had a different form?
- How does the unit step function impact the impulse response?
Tip: When dealing with sinusoidal inputs to LTI systems, analyzing the frequency response can often simplify finding the output amplitude.
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Math Problem Analysis
Mathematical Concepts
Linear Time-Invariant (LTI) Systems
Convolution
Fourier Transform
Amplitude Calculation
Formulas
Output \( y(t) = x(t) * h(t) \)
Fourier Transform of \( x(t) = 2\sin(t) \)
Fourier Transform of \( h(t) = e^{-t}u(t) \)
Magnitude of frequency response: \( |Y(\omega)| \)
Theorems
Frequency Response of LTI Systems
Suitable Grade Level
Undergraduate (Signal Processing or Electrical Engineering Courses)
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